Physlib.QuantumMechanics.HilbertSpaces.SpaceD.PolyBddSchwartzSubmodule

Physlib
QuantumMechanics
HilbertSpaces
SpaceD
PolyBddSchwartzSubmodule

Imports

/-
Copyright (c) 2026 Gregory J. Loges. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gregory J. Loges
-/
module

public import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
public import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
public import Physlib.QuantumMechanics.HilbertSpaces.SpaceD.SchwartzSubmodule

Polynomially-bounded Schwartz submodules

i. Overview

In this module we define polynomially-bounded Schwartz submodules of SpaceDHilbertSpace d μ.

For each a : ℕ∞, PolyBddSchwartzSubmodule d a μ is the submodule corresponding to Schwartz maps f satisfying the polynomial growth bounds ‖x‖ ^ (-k) * ‖f x‖ ≤ Cₖ for all (k : ℕ) ≤ a. In particular, for a = ⊤ such a bound holds for all natural numbers.

These serve as a natural domain for singular unbounded operators. For example, the 1/r Coulomb potential operator maps PolyBddSchwartzSubmodule d ⊤ μ to itself. In the same way that multiplying a Schwartz map by any polynomial in the coordinates results in a square-integrable function, polynomially-bounded Schwartz maps may be multiplied by Laurent polynomials and remain square-integrable (the precise condition depends on d, a and the negative degree of the Laurent polynomial).

Note: the condition defining polynomially-bounded Schwartz maps is phrased as ‖x‖ ^ (-k) * ‖f x‖ ≤ Cₖ rather than as ‖f x‖ ≤ Cₖ * ‖x‖ ^ k to mirror SchwartzMap.decay. These two conditions only differ at x = 0 and are therefore equivalent for d > 0 since then f 0 may be determined by continuity. For d = 0 the former does not constrain f 0 = 0 (since x = 0 is the only point and 0⁻¹ = 0) while the latter does (and would therefore spoil their being dense in SpaceDHilbertSpace 0 ≅ ℂ).

ii. Key results

PolyBddSchwartzSubmodule d (a : ℕ∞) μ: Restriction of SchwartzSubmodule d μ to those Schwartz maps which are bounded by powers of ‖x‖.

PolyBddSchwartzSubmodule.dense: These submodules are dense in SpaceDHilbertSpace d μ.

iii. Table of contents

A. Definitions

B. Coercions

C. (In)equalities

D. Density

iv. References

@[expose] public section

A. Definitions

A function is a bounded Schwartz map if it is both Schwartz and bounded by powers of ‖x‖.


def PolyBddSchwartzMap (d : ℕ) (a : ℕ∞) : Submodule ℂ 𝓢(Space d, ℂ) where
  carrier := {f : 𝓢(Space d, ℂ) |
    ∀ k : ℕ, k ≤ a → ∃ C : ℝ, 0 < C ∧ ∀ x : Space d, ‖x‖ ^ (-k : ℤ) * ‖f x‖ ≤ C}
  add_mem' := byd:ℕa:ℕ∞⊢ ∀ {a_1 b : 𝓢(Space d, ℂ)},
  a_1 ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C} →
    b ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C} →
      a_1 + b ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}
    intro f g hf hg k hkd:ℕa:ℕ∞f:𝓢(Space d, ℂ)g:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}hg:g ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ a⊢ ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖(f + g) x‖ ≤ C
    obtain ⟨C₁, hC₁_pos, hC₁⟩ := hf k hkd:ℕa:ℕ∞f:𝓢(Space d, ℂ)g:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}hg:g ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC₁:ℝhC₁_pos:0 < C₁hC₁:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C₁⊢ ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖(f + g) x‖ ≤ C
    obtain ⟨C₂, hC₂_pos, hC₂⟩ := hg k hkd:ℕa:ℕ∞f:𝓢(Space d, ℂ)g:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}hg:g ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC₁:ℝhC₁_pos:0 < C₁hC₁:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C₁C₂:ℝhC₂_pos:0 < C₂hC₂:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖g x‖ ≤ C₂⊢ ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖(f + g) x‖ ≤ C
    refine ⟨C₁ + C₂, byd:ℕa:ℕ∞f:𝓢(Space d, ℂ)g:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}hg:g ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC₁:ℝhC₁_pos:0 < C₁hC₁:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C₁C₂:ℝhC₂_pos:0 < C₂hC₂:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖g x‖ ≤ C₂⊢ 0 < C₁ + C₂ positivityAll goals completed! 🐙, fun x ↦ ?_⟩
    refine le_trans ?_ (add_le_add (hC₁ x) (hC₂ x))d:ℕa:ℕ∞f:𝓢(Space d, ℂ)g:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}hg:g ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC₁:ℝhC₁_pos:0 < C₁hC₁:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C₁C₂:ℝhC₂_pos:0 < C₂hC₂:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖g x‖ ≤ C₂x:Space d⊢ ‖x‖ ^ (-↑k) * ‖(f + g) x‖ ≤ ‖x‖ ^ (-↑k) * ‖f x‖ + ‖x‖ ^ (-↑k) * ‖g x‖
    rw [← mul_addd:ℕa:ℕ∞f:𝓢(Space d, ℂ)g:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}hg:g ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC₁:ℝhC₁_pos:0 < C₁hC₁:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C₁C₂:ℝhC₂_pos:0 < C₂hC₂:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖g x‖ ≤ C₂x:Space d⊢ ‖x‖ ^ (-↑k) * ‖(f + g) x‖ ≤ ‖x‖ ^ (-↑k) * (‖f x‖ + ‖g x‖)d:ℕa:ℕ∞f:𝓢(Space d, ℂ)g:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}hg:g ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC₁:ℝhC₁_pos:0 < C₁hC₁:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C₁C₂:ℝhC₂_pos:0 < C₂hC₂:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖g x‖ ≤ C₂x:Space d⊢ ‖x‖ ^ (-↑k) * ‖(f + g) x‖ ≤ ‖x‖ ^ (-↑k) * (‖f x‖ + ‖g x‖)]d:ℕa:ℕ∞f:𝓢(Space d, ℂ)g:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}hg:g ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC₁:ℝhC₁_pos:0 < C₁hC₁:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C₁C₂:ℝhC₂_pos:0 < C₂hC₂:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖g x‖ ≤ C₂x:Space d⊢ ‖x‖ ^ (-↑k) * ‖(f + g) x‖ ≤ ‖x‖ ^ (-↑k) * (‖f x‖ + ‖g x‖)
    exact mul_le_mul_of_nonneg_left (norm_add_le (f x) (g x)) (byd:ℕa:ℕ∞f:𝓢(Space d, ℂ)g:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}hg:g ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC₁:ℝhC₁_pos:0 < C₁hC₁:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C₁C₂:ℝhC₂_pos:0 < C₂hC₂:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖g x‖ ≤ C₂x:Space d⊢ 0 ≤ ‖x‖ ^ (-↑k) positivityAll goals completed! 🐙)
  zero_mem' := fun _ _ ↦ ⟨1, byd:ℕa:ℕ∞x✝¹:ℕx✝:↑x✝¹ ≤ a⊢ 0 < 1 ∧ ∀ (x : Space d), ‖x‖ ^ (-↑x✝¹) * ‖0 x‖ ≤ 1 simpAll goals completed! 🐙⟩
  smul_mem' := byd:ℕa:ℕ∞⊢ ∀ (c : ℂ) {x : 𝓢(Space d, ℂ)},
  x ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C} →
    c • x ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}
    intro c f hf k hkd:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ a⊢ ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖(c • f) x‖ ≤ C
    obtain ⟨C, hC_pos, hC⟩ := hf k hkd:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C⊢ ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖(c • f) x‖ ≤ C
    refine ⟨(1 + ‖c‖) * C, byd:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C⊢ 0 < (1 + ‖c‖) * C positivityAll goals completed! 🐙, fun x ↦ ?_⟩
    rw [smul_apply,d:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ Cx:Space d⊢ ‖x‖ ^ (-↑k) * ‖c • f x‖ ≤ (1 + ‖c‖) * Cd:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ Cx:Space d⊢ ‖c‖ * (‖x‖ ^ (-↑k) * ‖f x‖) ≤ (1 + ‖c‖) * C norm_smul,d:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ Cx:Space d⊢ ‖x‖ ^ (-↑k) * (‖c‖ * ‖f x‖) ≤ (1 + ‖c‖) * Cd:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ Cx:Space d⊢ ‖c‖ * (‖x‖ ^ (-↑k) * ‖f x‖) ≤ (1 + ‖c‖) * C mul_rotate',d:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ Cx:Space d⊢ ‖c‖ * (‖f x‖ * ‖x‖ ^ (-↑k)) ≤ (1 + ‖c‖) * Cd:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ Cx:Space d⊢ ‖c‖ * (‖x‖ ^ (-↑k) * ‖f x‖) ≤ (1 + ‖c‖) * C mul_comm ‖f x‖d:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ Cx:Space d⊢ ‖c‖ * (‖x‖ ^ (-↑k) * ‖f x‖) ≤ (1 + ‖c‖) * Cd:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ Cx:Space d⊢ ‖c‖ * (‖x‖ ^ (-↑k) * ‖f x‖) ≤ (1 + ‖c‖) * C]d:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ Cx:Space d⊢ ‖c‖ * (‖x‖ ^ (-↑k) * ‖f x‖) ≤ (1 + ‖c‖) * C
    exact le_trans (mul_le_mul_of_nonneg_left (hC x) (norm_nonneg c)) (byd:ℕa:ℕ∞c:ℂf:𝓢(Space d, ℂ)hf:f ∈ {f | ∀ (k : ℕ), ↑k ≤ a → ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C}k:ℕhk:↑k ≤ aC:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ Cx:Space d⊢ ‖c‖ * C ≤ (1 + ‖c‖) * C linarithAll goals completed! 🐙)

The continuous linear map schwartzIncl with domain restricted to PolyBddSchwartzMap d a.

def polyBddSchwartzIncl {d : ℕ} {a : ℕ∞} (μ : Measure (Space d)) [μ.HasTemperateGrowth] :
    PolyBddSchwartzMap d a →L[ℂ] SpaceDHilbertSpace d μ :=
  ⟨(schwartzIncl μ).domRestrict (PolyBddSchwartzMap d a),
    (schwartzIncl μ).continuous_domRestrict (schwartzIncl μ).continuous _⟩

The submodule of SpaceDHilbertSpace d corresponding to bounded Schwartz maps.

abbrev PolyBddSchwartzSubmodule
    (d : ℕ) (a : ℕ∞) (μ : Measure (Space d) := volume) [μ.HasTemperateGrowth] :
    Submodule ℂ (SpaceDHilbertSpace d μ) :=
  (polyBddSchwartzIncl (a := a) μ).range

lemma polyBddSchwartzIncl_injective
    {d : ℕ} (a : ℕ∞) (μ : Measure (Space d)) [μ.HasTemperateGrowth] [μ.IsOpenPosMeasure] :
    Function.Injective (polyBddSchwartzIncl (a := a) μ) :=
  LinearMap.injective_domRestrict_iff.mpr <| (schwartzIncl_ker μ).symm ▸ disjoint_bot_right

The linear equivalence between polynomially-bounded Schwartz maps and the corresponding submodule of the Hilbert space.

def polyBddSchwartzEquiv
    {d : ℕ} {a : ℕ∞} (μ : Measure (Space d)) [μ.HasTemperateGrowth] [μ.IsOpenPosMeasure] :
    PolyBddSchwartzMap d a ≃ₗ[ℂ] PolyBddSchwartzSubmodule d a μ :=
  LinearEquiv.ofInjective (polyBddSchwartzIncl μ).toLinearMap (polyBddSchwartzIncl_injective a μ)

B. Coercions

              instance : CoeOut (PolyBddSchwartzMap d a) 𝓢(Space d, ℂ) := ⟨fun f ↦ f.val⟩instance : CoeFun (PolyBddSchwartzMap d a) (fun _ ↦ Space d → ℂ) := ⟨fun f ↦ ⇑f.val⟩@[simp]
lemma toFun_eq_coe (f : PolyBddSchwartzMap d a) (x : Space d) : f.val.toFun x = f.val x := rfl

lemma polyBddSchwartzEquiv_symm_apply_coe [μ.IsOpenPosMeasure]
    {ψ : SchwartzSubmodule d μ} (hψ : ↑ψ ∈ PolyBddSchwartzSubmodule d a μ) :
    ((polyBddSchwartzEquiv μ).symm ⟨ψ, hψ⟩).val = (schwartzEquiv μ).symm ψ := byd:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μ⊢ ↑((polyBddSchwartzEquiv μ).symm ⟨↑ψ, hψ⟩) = (schwartzEquiv μ).symm ψ
  apply (schwartzEquiv μ).injectived:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μ⊢ (schwartzEquiv μ) ↑((polyBddSchwartzEquiv μ).symm ⟨↑ψ, hψ⟩) = (schwartzEquiv μ) ((schwartzEquiv μ).symm ψ)
  apply SetLike.coe_eq_coe.mpd:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μ⊢ ↑((schwartzEquiv μ) ↑((polyBddSchwartzEquiv μ).symm ⟨↑ψ, hψ⟩)) = ↑((schwartzEquiv μ) ((schwartzEquiv μ).symm ψ))
  obtain ⟨g, hg⟩ := (polyBddSchwartzEquiv μ).surjective ⟨ψ.val, hψ⟩d:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩⊢ ↑((schwartzEquiv μ) ↑((polyBddSchwartzEquiv μ).symm ⟨↑ψ, hψ⟩)) = ↑((schwartzEquiv μ) ((schwartzEquiv μ).symm ψ))
  have hg' : polyBddSchwartzIncl μ g = ψ := SetLike.coe_eq_coe.mpr hgd:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩hg':(polyBddSchwartzIncl μ) g = ↑ψ⊢ ↑((schwartzEquiv μ) ↑((polyBddSchwartzEquiv μ).symm ⟨↑ψ, hψ⟩)) = ↑((schwartzEquiv μ) ((schwartzEquiv μ).symm ψ))
  rw [← hg,d:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩hg':(polyBddSchwartzIncl μ) g = ↑ψ⊢ ↑((schwartzEquiv μ) ↑((polyBddSchwartzEquiv μ).symm ((polyBddSchwartzEquiv μ) g))) =
  ↑((schwartzEquiv μ) ((schwartzEquiv μ).symm ψ))d:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩hg':(polyBddSchwartzIncl μ) g = ↑ψ⊢ ↑((schwartzEquiv μ) ↑g) = (polyBddSchwartzIncl μ) g LinearEquiv.symm_apply_apply,d:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩hg':(polyBddSchwartzIncl μ) g = ↑ψ⊢ ↑((schwartzEquiv μ) ↑g) = ↑((schwartzEquiv μ) ((schwartzEquiv μ).symm ψ))d:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩hg':(polyBddSchwartzIncl μ) g = ↑ψ⊢ ↑((schwartzEquiv μ) ↑g) = (polyBddSchwartzIncl μ) g LinearEquiv.apply_symm_apply,d:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩hg':(polyBddSchwartzIncl μ) g = ↑ψ⊢ ↑((schwartzEquiv μ) ↑g) = ↑ψd:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩hg':(polyBddSchwartzIncl μ) g = ↑ψ⊢ ↑((schwartzEquiv μ) ↑g) = (polyBddSchwartzIncl μ) g ← hg'd:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩hg':(polyBddSchwartzIncl μ) g = ↑ψ⊢ ↑((schwartzEquiv μ) ↑g) = (polyBddSchwartzIncl μ) gd:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩hg':(polyBddSchwartzIncl μ) g = ↑ψ⊢ ↑((schwartzEquiv μ) ↑g) = (polyBddSchwartzIncl μ) g]d:ℕa:ℕ∞μ:Measure (Space d)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasureψ:↥(SchwartzSubmodule d μ)hψ:↑ψ ∈ PolyBddSchwartzSubmodule d a μg:↥(PolyBddSchwartzMap d a)hg:(polyBddSchwartzEquiv μ) g = ⟨↑ψ, hψ⟩hg':(polyBddSchwartzIncl μ) g = ↑ψ⊢ ↑((schwartzEquiv μ) ↑g) = (polyBddSchwartzIncl μ) g
  rflAll goals completed! 🐙

C. (In)equalities

              lemma PolyBddSchwartzMap_zero_eq_top (d : ℕ) : PolyBddSchwartzMap d 0 = ⊤ := byd:ℕ⊢ PolyBddSchwartzMap d 0 = ⊤
  ext fd:ℕf:𝓢(Space d, ℂ)⊢ f ∈ PolyBddSchwartzMap d 0 ↔ f ∈ ⊤
  have := f.decay 0 0d:ℕf:𝓢(Space d, ℂ)this:∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑f) x‖ ≤ C⊢ f ∈ PolyBddSchwartzMap d 0 ↔ f ∈ ⊤
  simp_all [PolyBddSchwartzMap]All goals completed! 🐙lemma PolyBddSchwartzMap_antitone (d : ℕ) {a b : ℕ∞} (h : a ≤ b) :
    PolyBddSchwartzMap d b ≤ PolyBddSchwartzMap d a := fun _ hx k hk ↦ hx k (hk.trans h)lemma of_zero_eq : PolyBddSchwartzSubmodule d 0 μ = SchwartzSubmodule d μ := byd:ℕμ:Measure (Space d)inst✝:μ.HasTemperateGrowth⊢ PolyBddSchwartzSubmodule d 0 μ = SchwartzSubmodule d μ
  simp [PolyBddSchwartzSubmodule, polyBddSchwartzIncl, PolyBddSchwartzMap_zero_eq_top]All goals completed! 🐙lemma le_SchwartzSubmodule (d : ℕ) (a : ℕ∞) : PolyBddSchwartzSubmodule d a ≤ SchwartzSubmodule d :=
  LinearMap.range_domRestrict_le_range _ _lemma antitone {a b : ℕ∞} (h : a ≤ b) :
    PolyBddSchwartzSubmodule d b μ ≤ PolyBddSchwartzSubmodule d a μ := byd:ℕμ:Measure (Space d)inst✝:μ.HasTemperateGrowtha:ℕ∞b:ℕ∞h:a ≤ b⊢ PolyBddSchwartzSubmodule d b μ ≤ PolyBddSchwartzSubmodule d a μ
  simp only [PolyBddSchwartzSubmodule, polyBddSchwartzIncl,
    ContinuousLinearMap.toLinearMap_domRestrict, LinearMap.range_domRestrict]d:ℕμ:Measure (Space d)inst✝:μ.HasTemperateGrowtha:ℕ∞b:ℕ∞h:a ≤ b⊢ Submodule.map (↑(schwartzIncl μ)) (PolyBddSchwartzMap d b) ≤ Submodule.map (↑(schwartzIncl μ)) (PolyBddSchwartzMap d a)
  exact Submodule.map_mono (PolyBddSchwartzMap_antitone d h)All goals completed! 🐙

D. Density

              

private lemma enorm_bump_mul_le_enorm {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E]
    [NormedSpace ℝ E] [HasContDiffBump E] {c : E} (f : ContDiffBump c) (g : E → 𝕜) (x : E) :
    ‖f x * g x‖ₑ ≤ ‖g x‖ₑ := by𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ ‖↑(↑f x) * g x‖ₑ ≤ ‖g x‖ₑ
  nth_rw 2 [← one_mul (g x)𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ ‖↑(↑f x) * g x‖ₑ ≤ ‖1 * g x‖ₑ]𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ ‖↑(↑f x) * g x‖ₑ ≤ ‖1 * g x‖ₑ
  simp_rw [𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ ‖↑(↑f x) * g x‖ₑ ≤ ‖1 * g x‖ₑenorm_mul𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ ‖↑(↑f x)‖ₑ * ‖g x‖ₑ ≤ ‖1‖ₑ * ‖g x‖ₑ]
  refine mul_le_mul_left ?_ ‖g x‖ₑ𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ ‖↑(↑f x)‖ₑ ≤ ‖1‖ₑ
  apply enorm_le_iff_norm_le.mpr𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ ‖↑(↑f x)‖ ≤ ‖1‖
  rw [norm_algebraMap',𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ ‖↑f x‖ ≤ ‖1‖𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ |↑f x| ≤ |1| Real.norm_eq_abs,𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ |↑f x| ≤ ‖1‖𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ |↑f x| ≤ |1| norm_one,𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ |↑f x| ≤ 1𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ |↑f x| ≤ |1| ← abs_one𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ |↑f x| ≤ |1|𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ |↑f x| ≤ |1|]𝕜:Type u_1E:Type u_2inst✝³:RCLike 𝕜inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:HasContDiffBump Ec:Ef:ContDiffBump cg:E → 𝕜x:E⊢ |↑f x| ≤ |1|
  exact abs_le_abs_of_nonneg f.nonneg f.le_oneAll goals completed! 🐙

private lemma dense_zero_top (μ : Measure (Space 0)) [μ.HasTemperateGrowth] [μ.IsOpenPosMeasure] :
    Dense (PolyBddSchwartzSubmodule 0 ⊤ μ : Set (SpaceDHilbertSpace 0 μ)) := byμ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasure⊢ Dense ↑(PolyBddSchwartzSubmodule 0 ⊤ μ)
  suffices PolyBddSchwartzMap 0 ⊤ = ⊤ byμ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasurethis:PolyBddSchwartzMap 0 ⊤ = ⊤⊢ Dense ↑(PolyBddSchwartzSubmodule 0 ⊤ μ)μ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasure⊢ PolyBddSchwartzMap 0 ⊤ = ⊤
    simp [PolyBddSchwartzSubmodule, polyBddSchwartzIncl, this]μ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasure⊢ PolyBddSchwartzMap 0 ⊤ = ⊤μ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasure⊢ PolyBddSchwartzMap 0 ⊤ = ⊤
  refine Submodule.eq_top_iff'.mpr (fun f k hk ↦ ?_)μ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasuref:𝓢(Space 0, ℂ)k:ℕhk:↑k ≤ ⊤⊢ ∃ C, 0 < C ∧ ∀ (x : Space 0), ‖x‖ ^ (-↑k) * ‖f x‖ ≤ C
  refine ⟨1 + ‖f 0‖, byμ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasuref:𝓢(Space 0, ℂ)k:ℕhk:↑k ≤ ⊤⊢ 0 < 1 + ‖f 0‖ positivityAll goals completed! 🐙, fun x ↦ ?_⟩
  simp only [Space.point_dim_zero_eq, norm_zero, zpow_neg, zpow_natCast]μ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasuref:𝓢(Space 0, ℂ)k:ℕhk:↑k ≤ ⊤x:Space 0⊢ (0 ^ k)⁻¹ * ‖f 0‖ ≤ 1 + ‖f 0‖
  rcases k with _ | kzeroμ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasuref:𝓢(Space 0, ℂ)x:Space 0hk:↑0 ≤ ⊤⊢ (0 ^ 0)⁻¹ * ‖f 0‖ ≤ 1 + ‖f 0‖succμ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasuref:𝓢(Space 0, ℂ)x:Space 0k:ℕhk:↑(k + 1) ≤ ⊤⊢ (0 ^ (k + 1))⁻¹ * ‖f 0‖ ≤ 1 + ‖f 0‖ <;>zeroμ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasuref:𝓢(Space 0, ℂ)x:Space 0hk:↑0 ≤ ⊤⊢ (0 ^ 0)⁻¹ * ‖f 0‖ ≤ 1 + ‖f 0‖succμ:Measure (Space 0)inst✝¹:μ.HasTemperateGrowthinst✝:μ.IsOpenPosMeasuref:𝓢(Space 0, ℂ)x:Space 0k:ℕhk:↑(k + 1) ≤ ⊤⊢ (0 ^ (k + 1))⁻¹ * ‖f 0‖ ≤ 1 + ‖f 0‖ simp [add_nonneg]All goals completed! 🐙TODO "Generalize density of PolyBddSchwartzSubmodule to more general measures than just μ ≤ volume."

lemma dense_top (hμ : μ ≤ volume) [μ.IsOpenPosMeasure] [IsFiniteMeasureOnCompacts μ] :
    Dense (PolyBddSchwartzSubmodule d ⊤ μ : Set (SpaceDHilbertSpace d μ)) := byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μ⊢ Dense ↑(PolyBddSchwartzSubmodule d ⊤ μ)
  rcases eq_zero_or_pos d with (rfl | hd)inlμ:Measure (Space 0)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μ⊢ Dense ↑(PolyBddSchwartzSubmodule 0 ⊤ μ)inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < d⊢ Dense ↑(PolyBddSchwartzSubmodule d ⊤ μ)
  ·inlμ:Measure (Space 0)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μ⊢ Dense ↑(PolyBddSchwartzSubmodule 0 ⊤ μ) -- `d = 0`: Every function `Space 0 ≅ {0} → ℂ` is in `PolyBddSchwartzSubmodule 0 ⊤ μ`.
    exact dense_zero_top _All goals completed! 🐙
  ·inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < d⊢ Dense ↑(PolyBddSchwartzSubmodule d ⊤ μ) -- `d > 0`: Construct a sequence in `PolyBddSchwartzSubmodule d ⊤ μ` which tends to `ξ`
    intro ξinrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)⊢ ξ ∈ closure ↑(PolyBddSchwartzSubmodule d ⊤ μ)
    apply mem_closure_iff_seq_limit.mprinrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)
    -- `ψₙ = [fₙ]` is a sequence in `SchwartzSubmodule` which tends to `ξ`
    obtain ⟨ψ, hψ, hψξ⟩ := mem_closure_iff_seq_limit.mp (SchwartzSubmodule.dense d μ ξ)inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)
    let f (n : ℕ) : 𝓢(Space d, ℂ) := (schwartzEquiv μ).symm ⟨ψ n, hψ n⟩inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)
    -- `bₙ` is a sequence of bump functions with shrinking domain
    let b (n : ℕ) : ContDiffBump (0 : Space d) :=
      ⟨(n + 1)⁻¹, 2 * (n + 1 : ℝ)⁻¹, byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩n:ℕ⊢ 0 < (↑n + 1)⁻¹inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ) positivityAll goals completed! 🐙inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ), lt_two_mul_self Nat.inv_pos_of_nat⟩inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)
    -- `φₙ = [bₙfₙ]` is a sequence in `SchwartzSubmodule` which tends to `0`
    let g (n : ℕ) : 𝓢(Space d, ℂ) := smulLeftCLM ℂ (b n) (f n)inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)
    let φ (n : ℕ) : SpaceDHilbertSpace d μ := schwartzIncl μ (g n)inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)
    have hg (n : ℕ) (x : Space d) : g n x = b n x * f n x := byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μ⊢ Dense ↑(PolyBddSchwartzSubmodule d ⊤ μ)inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)
      have := (b n).hasCompactSupport.hasTemperateGrowth (b n).contDiffd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)n:ℕx:Space dthis:Function.HasTemperateGrowth ↑(b n)⊢ (g n) x = ↑(↑(b n) x) * (f n) xinrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)
      rw [smulLeftCLM_apply_apply this,d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)n:ℕx:Space dthis:Function.HasTemperateGrowth ↑(b n)⊢ ↑(b n) x • (f n) x = ↑(↑(b n) x) * (f n) xinrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ) ← Complex.coe_smul,d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)n:ℕx:Space dthis:Function.HasTemperateGrowth ↑(b n)⊢ ↑(↑(b n) x) • (f n) x = ↑(↑(b n) x) * (f n) xinrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ) smul_eq_muld:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)n:ℕx:Space dthis:Function.HasTemperateGrowth ↑(b n)⊢ ↑(↑(b n) x) * (f n) x = ↑(↑(b n) x) * (f n) xinrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)]inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ ∃ x, (∀ (n : ℕ), x n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto x atTop (nhds ξ)
    use ψ - φhd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ (∀ (n : ℕ), (ψ - φ) n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)) ∧ Tendsto (ψ - φ) atTop (nhds ξ)
    constructorh.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ ∀ (n : ℕ), (ψ - φ) n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto (ψ - φ) atTop (nhds ξ)
    ·h.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ ∀ (n : ℕ), (ψ - φ) n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ) intro nh.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕ⊢ (ψ - φ) n ∈ ↑(PolyBddSchwartzSubmodule d ⊤ μ)
      rw [SetLike.mem_coe,h.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕ⊢ (ψ - φ) n ∈ PolyBddSchwartzSubmodule d ⊤ μh.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕ⊢ ∃ a, ∃ (b : a ∈ PolyBddSchwartzMap d ⊤), ↑(polyBddSchwartzIncl μ) ⟨a, b⟩ = (ψ - φ) n LinearMap.mem_range,h.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕ⊢ ∃ y, ↑(polyBddSchwartzIncl μ) y = (ψ - φ) nh.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕ⊢ ∃ a, ∃ (b : a ∈ PolyBddSchwartzMap d ⊤), ↑(polyBddSchwartzIncl μ) ⟨a, b⟩ = (ψ - φ) n Subtype.existsh.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕ⊢ ∃ a, ∃ (b : a ∈ PolyBddSchwartzMap d ⊤), ↑(polyBddSchwartzIncl μ) ⟨a, b⟩ = (ψ - φ) nh.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕ⊢ ∃ a, ∃ (b : a ∈ PolyBddSchwartzMap d ⊤), ↑(polyBddSchwartzIncl μ) ⟨a, b⟩ = (ψ - φ) n]h.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕ⊢ ∃ a, ∃ (b : a ∈ PolyBddSchwartzMap d ⊤), ↑(polyBddSchwartzIncl μ) ⟨a, b⟩ = (ψ - φ) n
      refine ⟨f n - g n, ?_, byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕ⊢ ↑(polyBddSchwartzIncl μ) ⟨f n - g n, ?m.288⟩ = (ψ - φ) n simp [f, φ, polyBddSchwartzIncl, ← schwartzEquiv_apply_coe]All goals completed! 🐙⟩
      intro k _h.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤⊢ ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖(f n - g n) x‖ ≤ C
      obtain ⟨C, hC_pos, hC⟩ := (f n).decay 0 0h.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑(f n)) x‖ ≤ C⊢ ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖(f n - g n) x‖ ≤ C
      simp only [pow_zero, norm_iteratedFDeriv_zero, one_mul] at hCh.leftd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ C⊢ ∃ C, 0 < C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖(f n - g n) x‖ ≤ C
      use (n + 1) ^ k * Chd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ C⊢ 0 < (↑n + 1) ^ k * C ∧ ∀ (x : Space d), ‖x‖ ^ (-↑k) * ‖(f n - g n) x‖ ≤ (↑n + 1) ^ k * C
      refine ⟨byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ C⊢ 0 < (↑n + 1) ^ k * C positivityAll goals completed! 🐙, fun x ↦ ?_⟩
      rcases le_or_gt ‖x‖ (b n).rIn with (hx | hx)h.inld:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:‖x‖ ≤ (b n).rIn⊢ ‖x‖ ^ (-↑k) * ‖(f n - g n) x‖ ≤ (↑n + 1) ^ k * Ch.inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖x‖ ^ (-↑k) * ‖(f n - g n) x‖ ≤ (↑n + 1) ^ k * C
      ·h.inld:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:‖x‖ ≤ (b n).rIn⊢ ‖x‖ ^ (-↑k) * ‖(f n - g n) x‖ ≤ (↑n + 1) ^ k * C have h_one : b n x = 1 := (b n).one_of_mem_closedBall (mem_closedBall_zero_iff.mpr hx)h.inld:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:‖x‖ ≤ (b n).rInh_one:↑(b n) x = 1⊢ ‖x‖ ^ (-↑k) * ‖(f n - g n) x‖ ≤ (↑n + 1) ^ k * C
        exact le_of_eq_of_le (b := 0) (byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:‖x‖ ≤ (b n).rInh_one:↑(b n) x = 1⊢ ‖x‖ ^ (-↑k) * ‖(f n - g n) x‖ = 0 simp [hg, h_one]All goals completed! 🐙) (byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:‖x‖ ≤ (b n).rInh_one:↑(b n) x = 1⊢ 0 ≤ (↑n + 1) ^ k * C positivityAll goals completed! 🐙)
      ·h.inrd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖x‖ ^ (-↑k) * ‖(f n - g n) x‖ ≤ (↑n + 1) ^ k * C refine mul_le_mul_of_nonneg ?_ ?_ (byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ 0 ≤ ‖x‖ ^ (-↑k) positivityAll goals completed! 🐙) hC_pos.le
        ·h.inr.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖x‖ ^ (-↑k) ≤ (↑n + 1) ^ k rw [← inv_zpow',h.inr.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖x‖⁻¹ ^ ↑k ≤ (↑n + 1) ^ kh.inr.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖x‖⁻¹ ^ k ≤ (↑n + 1) ^ k zpow_natCasth.inr.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖x‖⁻¹ ^ k ≤ (↑n + 1) ^ kh.inr.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖x‖⁻¹ ^ k ≤ (↑n + 1) ^ k]h.inr.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖x‖⁻¹ ^ k ≤ (↑n + 1) ^ k
          gcongrhabd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖x‖⁻¹ ≤ ↑n + 1
          exact (inv_lt_of_inv_lt₀ (Nat.cast_add_one_pos n) hx).leAll goals completed! 🐙
        ·h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖(f n - g n) x‖ ≤ C refine le_trans ?_ (hC x)h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖(f n - g n) x‖ ≤ ‖(f n) x‖
          rw [sub_apply,h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖(f n) x - (g n) x‖ ≤ ‖(f n) x‖h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ 1 * ‖(f n) x‖ ← one_mul (f n x),h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 * (f n) x - (g n) x‖ ≤ ‖1 * (f n) x‖h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ 1 * ‖(f n) x‖ hg,h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 * (f n) x - ↑(↑(b n) x) * (f n) x‖ ≤ ‖1 * (f n) x‖h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ 1 * ‖(f n) x‖ ← sub_mul,h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖(1 - ↑(↑(b n) x)) * (f n) x‖ ≤ ‖1 * (f n) x‖h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ 1 * ‖(f n) x‖ norm_mul,h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ ‖1 * (f n) x‖h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ 1 * ‖(f n) x‖ norm_mul,h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ ‖1‖ * ‖(f n) x‖h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ 1 * ‖(f n) x‖ norm_oneh.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ 1 * ‖(f n) x‖h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ 1 * ‖(f n) x‖]h.inr.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ * ‖(f n) x‖ ≤ 1 * ‖(f n) x‖
          gcongrhbcd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(↑(b n) x)‖ ≤ 1
          rw [← ofReal_one,hbcd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖↑1 - ↑(↑(b n) x)‖ ≤ 1hbcd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ |1 - ↑(b n) x| ≤ 1 ← ofReal_sub,hbcd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖↑(1 - ↑(b n) x)‖ ≤ 1hbcd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ |1 - ↑(b n) x| ≤ 1 norm_real,hbcd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ ‖1 - ↑(b n) x‖ ≤ 1hbcd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ |1 - ↑(b n) x| ≤ 1 Real.norm_eq_abshbcd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ |1 - ↑(b n) x| ≤ 1hbcd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ |1 - ↑(b n) x| ≤ 1]hbcd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xn:ℕk:ℕa✝:↑k ≤ ⊤C:ℝhC_pos:0 < ChC:∀ (x : Space d), ‖(f n) x‖ ≤ Cx:Space dhx:(b n).rIn < ‖x‖⊢ |1 - ↑(b n) x| ≤ 1
          exact abs_sub_le_of_nonneg_of_le zero_le_one le_rfl (b n).nonneg (b n).le_oneAll goals completed! 🐙
    ·h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto (ψ - φ) atTop (nhds ξ) refine tendsto_of_sub_tendsto_zero ξ hψξ ?_h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto (ψ - φ - ψ) atTop (nhds 0)
      rw [sub_sub_cancel_left,h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto (-φ) atTop (nhds 0)h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto φ atTop (nhds 0) Pi.neg_def,h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto (fun i => -φ i) atTop (nhds 0)h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto φ atTop (nhds 0) ← neg_zero,h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto (fun i => -φ i) atTop (nhds (-0))h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto φ atTop (nhds 0) tendsto_neg_iffh.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto φ atTop (nhds 0)h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto φ atTop (nhds 0)]h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) x⊢ Tendsto φ atTop (nhds 0)
      -- Split `φₙ = σₙ + (φₙ - σₐ)` with `σₙ ≔ [bₙξ]` a sequence in `SpaceDHilbertSpace`
      let s (n : ℕ) : Space d → ℂ := fun x ↦ b n x * ξ xh.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ x⊢ Tendsto φ atTop (nhds 0)
      let σ (n : ℕ) : SpaceDHilbertSpace d μ := byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xn:ℕ⊢ ↥(SpaceDHilbertSpace d μ)h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯⊢ Tendsto φ atTop (nhds 0)
        refine mk (f := s n) ⟨?_, ?_⟩refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xn:ℕ⊢ AEStronglyMeasurable (s n) μrefine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xn:ℕ⊢ eLpNorm (s n) 2 μ < ⊤h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯⊢ Tendsto φ atTop (nhds 0)
        ·refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xn:ℕ⊢ AEStronglyMeasurable (s n) μh.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯⊢ Tendsto φ atTop (nhds 0) exact (continuous_ofReal.comp (b n).continuous).aestronglyMeasurable.mul
            ξ.val.aestronglyMeasurableAll goals completed! 🐙h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯⊢ Tendsto φ atTop (nhds 0)
        ·refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xn:ℕ⊢ eLpNorm (s n) 2 μ < ⊤h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯⊢ Tendsto φ atTop (nhds 0) refine lt_of_le_of_lt ?_ (memHS_coe ξ).2refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xn:ℕ⊢ eLpNorm (s n) 2 μ ≤ eLpNorm (↑↑ξ) 2 μh.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯⊢ Tendsto φ atTop (nhds 0)
          exact eLpNorm_mono_enorm (enorm_bump_mul_le_enorm (b n) ξ)h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯⊢ Tendsto φ atTop (nhds 0)h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯⊢ Tendsto φ atTop (nhds 0)
      have hψ_ae (n : ℕ) : ψ n =ᵐ[μ] f n := (schwartzEquiv_symm_coe_ae ⟨ψ n, hψ n⟩).symmh.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)⊢ Tendsto φ atTop (nhds 0)
      have hφ_ae (n : ℕ) : φ n =ᵐ[μ] g n := schwartzEquiv_coe_ae (g n)h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)⊢ Tendsto φ atTop (nhds 0)
      have hσ_ae (n : ℕ) : σ n =ᵐ[μ] s n := coeFn_mk _h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s n⊢ Tendsto φ atTop (nhds 0)
      have hφσ_ae (n : ℕ) : (φ - σ) n =ᵐ[μ] g n - s n :=
        (coeFn_sub (φ n) (σ n)).trans <| (hφ_ae n).sub (hσ_ae n)h.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s n⊢ Tendsto φ atTop (nhds 0)
      have hψξ_ae (n : ℕ) : ψ n - ξ =ᵐ[μ] f n - ξ :=
        (coeFn_sub (ψ n) ξ).trans <| (hψ_ae n).sub EventuallyEq.rflh.rightd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξ⊢ Tendsto φ atTop (nhds 0)
      refine tendsto_of_sub_tendsto_zero (f := σ) 0 ?_ ?_h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξ⊢ Tendsto σ atTop (nhds 0)h.right.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξ⊢ Tendsto (φ - σ) atTop (nhds 0)
      ·h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξ⊢ Tendsto σ atTop (nhds 0) -- `σ = bₙξ → 0` since the norms are bounded by the integral of `‖ξ‖²` (independent of `n`!)
        -- on a domain which tends to zero
        apply tendsto_zero_iff_tendsto_zero_lintegral_enorm_sq.mprh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξ⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
        let B (n : ℕ) : Set (Space d) := Metric.ball 0 (b n).rOuth.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOut⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
        have hξB : Tendsto (fun n ↦ ∫⁻ x in B n, ‖ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0) := byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μ⊢ Dense ↑(PolyBddSchwartzSubmodule d ⊤ μ)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
          refine tendsto_setLIntegral_zero ?_ ?_refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOut⊢ ∫⁻ (x : Space d), ‖↑↑ξ x‖ₑ ^ 2 ∂μ ≠ ⊤refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOut⊢ Tendsto (⇑μ ∘ B) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
          ·refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOut⊢ ∫⁻ (x : Space d), ‖↑↑ξ x‖ₑ ^ 2 ∂μ ≠ ⊤h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0) refine lt_top_iff_ne_top.mp ?_refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOut⊢ ∫⁻ (x : Space d), ‖↑↑ξ x‖ₑ ^ 2 ∂μ < ⊤h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
            simpa [eLpNorm_one_eq_lintegral_enorm, Real.rpow_ofNat, enorm_pow, enorm_norm]
              using L2.eLpNorm_rpow_two_norm_lt_top ξAll goals completed! 🐙h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
          ·refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOut⊢ Tendsto (⇑μ ∘ B) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0) have : NeZero d := ⟨hd.ne'⟩refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero d⊢ Tendsto (⇑μ ∘ B) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
            refine tendsto_const_nhds.squeeze ?_ zero_le (fun n ↦ hμ (B n))refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero d⊢ Tendsto (fun n => volume (B n)) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
            let C : ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (d / 2 + 1))).toRealrefine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toReal⊢ Tendsto (fun n => volume (B n)) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
            have hvolB : ∀ n, volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d) := byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μ⊢ Dense ↑(PolyBddSchwartzSubmodule d ⊤ μ)refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => volume (B n)) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
              intro nd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealn:ℕ⊢ volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => volume (B n)) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
              simp [B, InnerProductSpace.volume_ball, C, mul_comm,
                ENNReal.ofReal_pow (b n).rOut_pos.le]refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => volume (B n)) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => volume (B n)) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
            simp_rw [refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => volume (B n)) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)hvolB,refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * (b n).rOut ^ d)) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0) ← ENNReal.ofReal_zero,refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * (b n).rOut ^ d)) atTop (nhds (ENNReal.ofReal 0))h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0) b,refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * (2 * (↑n + 1)⁻¹) ^ d)) atTop (nhds (ENNReal.ofReal 0))h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0) ← one_div,refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * (2 * (1 / (↑n + 1))) ^ d)) atTop (nhds (ENNReal.ofReal 0))h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0) mul_pow,refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * (2 ^ d * (1 / (↑n + 1)) ^ d))) atTop (nhds (ENNReal.ofReal 0))h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0) ← mul_assocrefine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * 2 ^ d * (1 / (↑n + 1)) ^ d)) atTop (nhds (ENNReal.ofReal 0))h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)]
            rw [← mul_zero (C * 2 ^ d),refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * 2 ^ d * (1 / (↑n + 1)) ^ d)) atTop (nhds (ENNReal.ofReal (C * 2 ^ d * 0)))refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * 2 ^ d * (1 / (↑n + 1)) ^ d)) atTop (nhds (ENNReal.ofReal (C * 2 ^ d * 0 ^ d)))h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0) ← zero_pow (M₀ := ℝ) hd.ne'refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * 2 ^ d * (1 / (↑n + 1)) ^ d)) atTop (nhds (ENNReal.ofReal (C * 2 ^ d * 0 ^ d)))refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * 2 ^ d * (1 / (↑n + 1)) ^ d)) atTop (nhds (ENNReal.ofReal (C * 2 ^ d * 0 ^ d)))h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)]refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => ENNReal.ofReal (C * 2 ^ d * (1 / (↑n + 1)) ^ d)) atTop (nhds (ENNReal.ofReal (C * 2 ^ d * 0 ^ d)))h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
            refine ENNReal.tendsto_ofReal <| Tendsto.const_mul (C * 2 ^ d) ?_refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOutthis:NeZero dC:ℝ := (ENNReal.ofReal (√Real.pi ^ d / Real.Gamma (↑d / 2 + 1))).toRealhvolB:∀ (n : ℕ), volume (B n) = ENNReal.ofReal (C * (b n).rOut ^ d)⊢ Tendsto (fun n => (1 / (↑n + 1)) ^ d) atTop (nhds (0 ^ d))h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
            exact tendsto_one_div_add_atTop_nhds_zero_nat.pow dh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑(σ a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
        refine tendsto_const_nhds.squeeze hξB (zero_le) (fun n ↦ ?_)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ ≤ ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ
        suffices ∫⁻ x, ‖σ n x‖ₑ ^ 2 ∂μ = ∫⁻ x in B n, ‖σ n x‖ₑ ^ 2 ∂μ byd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕthis:∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ ≤ ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ
          rw [thisd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕthis:∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ ≤ ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕthis:∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ ≤ ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ]d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕthis:∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ ≤ ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ
          refine setLIntegral_mono_ae' measurableSet_ball ?_d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕthis:∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ⊢ ∀ᵐ (x : Space d) ∂μ, x ∈ B n → ‖↑↑(σ n) x‖ₑ ^ 2 ≤ ‖↑↑ξ x‖ₑ ^ 2h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ
          filter_upwards [hσ_ae n] with x hd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕthis:∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μx:Space dh:↑↑(σ n) x = s n x⊢ x ∈ B n → ‖↑↑(σ n) x‖ₑ ^ 2 ≤ ‖↑↑ξ x‖ₑ ^ 2h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ _d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕthis:∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μx:Space dh:↑↑(σ n) x = s n xa✝:x ∈ B n⊢ ‖↑↑(σ n) x‖ₑ ^ 2 ≤ ‖↑↑ξ x‖ₑ ^ 2h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ
          exact ENNReal.pow_le_pow_left <| h ▸ enorm_bump_mul_le_enorm (b n) ξ xh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ
        have h (A : Set (Space d)) : ∫⁻ x in A, ‖σ n x‖ₑ ^ 2 ∂μ = ∫⁻ x in A, ‖s n x‖ₑ ^ 2 ∂μ :=
          lintegral_congr_ae ((hσ_ae n).fun_comp (fun z ↦ ‖z‖ₑ ^ 2)).restricth.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d), ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ
        rw [← setLIntegral_univ,h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d) in Set.univ, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d), ‖s n x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖s n x‖ₑ ^ 2 ∂μ h,h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d) in Set.univ, ‖s n x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d), ‖s n x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖s n x‖ₑ ^ 2 ∂μ h,h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d) in Set.univ, ‖s n x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖s n x‖ₑ ^ 2 ∂μh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d), ‖s n x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖s n x‖ₑ ^ 2 ∂μ setLIntegral_univh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d), ‖s n x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖s n x‖ₑ ^ 2 ∂μh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d), ‖s n x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖s n x‖ₑ ^ 2 ∂μ]h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ ∫⁻ (x : Space d), ‖s n x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in B n, ‖s n x‖ₑ ^ 2 ∂μ
        refine (setLIntegral_eq_of_support_subset ?_).symmh.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μ⊢ (Function.support fun x => ‖s n x‖ₑ ^ 2) ⊆ B n
        refine Function.support_subset_iff'.mpr (fun x hx ↦ ?_)h.right.refine_1d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξB:ℕ → Set (Space d) := fun n => Metric.ball 0 (b n).rOuthξB:Tendsto (fun n => ∫⁻ (x : Space d) in B n, ‖↑↑ξ x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕh:∀ (A : Set (Space d)), ∫⁻ (x : Space d) in A, ‖↑↑(σ n) x‖ₑ ^ 2 ∂μ = ∫⁻ (x : Space d) in A, ‖s n x‖ₑ ^ 2 ∂μx:Space dhx:x ∉ B n⊢ ‖s n x‖ₑ ^ 2 = 0
        simp [s, (b n).zero_of_le_dist (not_lt.mp hx)]All goals completed! 🐙
      ·h.right.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξ⊢ Tendsto (φ - σ) atTop (nhds 0) -- `φₙ - σₙ = bₙ(ψₙ - ξ) → 0` since `ψₙ → ξ` (by definition) and the `bₙ` are bounded
        apply tendsto_zero_iff_tendsto_zero_lintegral_enorm_sq.mprh.right.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξ⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑((φ - σ) a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
        have hψξ : Tendsto (fun n ↦ ∫⁻ x, ‖(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0) :=
          tendsto_zero_iff_tendsto_zero_lintegral_enorm_sq.mp (sub_self ξ ▸ hψξ.sub_const ξ)h.right.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)⊢ Tendsto (fun a => ∫⁻ (x : Space d), ‖↑↑((φ - σ) a) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)
        refine Tendsto.squeeze tendsto_const_nhds hψξ (zero_le) (fun n ↦ ?_)h.right.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∫⁻ (x : Space d), ‖↑↑((φ - σ) n) x‖ₑ ^ 2 ∂μ ≤ ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ
        refine lintegral_mono_ae ?_h.right.refine_2d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕ⊢ ∀ᵐ (a : Space d) ∂μ, ‖↑↑((φ - σ) n) a‖ₑ ^ 2 ≤ ‖↑↑(ψ n - ξ) a‖ₑ ^ 2
        filter_upwards [hφσ_ae n, hψξ_ae n] with x hd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕx:Space dh:↑↑((φ - σ) n) x = (⇑(g n) - s n) x⊢ ↑↑(ψ n - ξ) x = (⇑(f n) - ↑↑ξ) x → ‖↑↑((φ - σ) n) x‖ₑ ^ 2 ≤ ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 h'd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕx:Space dh:↑↑((φ - σ) n) x = (⇑(g n) - s n) xh':↑↑(ψ n - ξ) x = (⇑(f n) - ↑↑ξ) x⊢ ‖↑↑((φ - σ) n) x‖ₑ ^ 2 ≤ ‖↑↑(ψ n - ξ) x‖ₑ ^ 2
        simp_rw [d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕx:Space dh:↑↑((φ - σ) n) x = (⇑(g n) - s n) xh':↑↑(ψ n - ξ) x = (⇑(f n) - ↑↑ξ) x⊢ ‖↑↑((φ - σ) n) x‖ₑ ^ 2 ≤ ‖↑↑(ψ n - ξ) x‖ₑ ^ 2h,d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕx:Space dh:↑↑((φ - σ) n) x = (⇑(g n) - s n) xh':↑↑(ψ n - ξ) x = (⇑(f n) - ↑↑ξ) x⊢ ‖(⇑(g n) - s n) x‖ₑ ^ 2 ≤ ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 h',d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕx:Space dh:↑↑((φ - σ) n) x = (⇑(g n) - s n) xh':↑↑(ψ n - ξ) x = (⇑(f n) - ↑↑ξ) x⊢ ‖(⇑(g n) - s n) x‖ₑ ^ 2 ≤ ‖(⇑(f n) - ↑↑ξ) x‖ₑ ^ 2 Pi.sub_apply,d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕx:Space dh:↑↑((φ - σ) n) x = (⇑(g n) - s n) xh':↑↑(ψ n - ξ) x = (⇑(f n) - ↑↑ξ) x⊢ ‖(g n) x - s n x‖ₑ ^ 2 ≤ ‖(f n) x - ↑↑ξ x‖ₑ ^ 2 hg,d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕx:Space dh:↑↑((φ - σ) n) x = (⇑(g n) - s n) xh':↑↑(ψ n - ξ) x = (⇑(f n) - ↑↑ξ) x⊢ ‖↑(↑(b n) x) * (f n) x - s n x‖ₑ ^ 2 ≤ ‖(f n) x - ↑↑ξ x‖ₑ ^ 2 s,d:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕx:Space dh:↑↑((φ - σ) n) x = (⇑(g n) - s n) xh':↑↑(ψ n - ξ) x = (⇑(f n) - ↑↑ξ) x⊢ ‖↑(↑(b n) x) * (f n) x - ↑(↑(b n) x) * ↑↑ξ x‖ₑ ^ 2 ≤ ‖(f n) x - ↑↑ξ x‖ₑ ^ 2 ← mul_subd:ℕμ:Measure (Space d)inst✝²:μ.HasTemperateGrowthhμ:μ ≤ volumeinst✝¹:μ.IsOpenPosMeasureinst✝:IsFiniteMeasureOnCompacts μhd:0 < dξ:↥(SpaceDHilbertSpace d μ)ψ:ℕ → ↥(SpaceDHilbertSpace d μ)hψ:∀ (n : ℕ), ψ n ∈ ↑(SchwartzSubmodule d μ)hψξ✝:Tendsto ψ atTop (nhds ξ)f:ℕ → 𝓢(Space d, ℂ) := fun n => (schwartzEquiv μ).symm ⟨ψ n, ⋯⟩b:ℕ → ContDiffBump 0 := fun n => { rIn := (↑n + 1)⁻¹, rOut := 2 * (↑n + 1)⁻¹, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }g:ℕ → 𝓢(Space d, ℂ) := fun n => (smulLeftCLM ℂ ↑(b n)) (f n)φ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => (schwartzIncl μ) (g n)hg:∀ (n : ℕ) (x : Space d), (g n) x = ↑(↑(b n) x) * (f n) xs:ℕ → Space d → ℂ := fun n x => ↑(↑(b n) x) * ↑↑ξ xσ:ℕ → ↥(SpaceDHilbertSpace d μ) := fun n => mk ⋯hψ_ae:∀ (n : ℕ), ↑↑(ψ n) =ᵐ[μ] ⇑(f n)hφ_ae:∀ (n : ℕ), ↑↑(φ n) =ᵐ[μ] ⇑(g n)hσ_ae:∀ (n : ℕ), ↑↑(σ n) =ᵐ[μ] s nhφσ_ae:∀ (n : ℕ), ↑↑((φ - σ) n) =ᵐ[μ] ⇑(g n) - s nhψξ_ae:∀ (n : ℕ), ↑↑(ψ n - ξ) =ᵐ[μ] ⇑(f n) - ↑↑ξhψξ:Tendsto (fun n => ∫⁻ (x : Space d), ‖↑↑(ψ n - ξ) x‖ₑ ^ 2 ∂μ) atTop (nhds 0)n:ℕx:Space dh:↑↑((φ - σ) n) x = (⇑(g n) - s n) xh':↑↑(ψ n - ξ) x = (⇑(f n) - ↑↑ξ) x⊢ ‖↑(↑(b n) x) * ((f n) x - ↑↑ξ x)‖ₑ ^ 2 ≤ ‖(f n) x - ↑↑ξ x‖ₑ ^ 2]
        exact ENNReal.pow_le_pow_left <| enorm_bump_mul_le_enorm (b n) (fun x ↦ f n x - ξ x) xAll goals completed! 🐙lemma dense (hμ : μ ≤ volume) [μ.IsOpenPosMeasure] [IsFiniteMeasureOnCompacts μ] :
    Dense (PolyBddSchwartzSubmodule d a μ : Set (SpaceDHilbertSpace d μ)) :=
  (dense_top μ hμ).mono (antitone μ le_top)