Module: Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff

Goal state before:
pp: E : Type u_1
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
inst✝¹¹ : FiniteDimensional ℝ E
F : Type u_2
inst✝¹⁰ : NormedAddCommGroup F
inst✝⁹ : NormedSpace ℝ F
inst✝⁸ : CompleteSpace F
H : Type u_3
inst✝⁷ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type u_4
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace H M
inst✝⁴ : IsManifold I ∞ M
inst✝³ : MeasurableSpace M
inst✝² : BorelSpace M
inst✝¹ : T2Space M
f : M → F
μ : Measure M
inst✝ : SigmaCompactSpace M
hf : LocallyIntegrable f μ
h : ∀ (g : M → ℝ), ContMDiff I 𝓘(ℝ, ℝ) ∞ g → HasCompactSupport g → ∫ (x : M), g x • f x ∂μ = 0
⊢ ∀ᵐ (x : M) ∂μ, f x = 0

Tactic:
by
  -- record topological properties of `M`
  have := I.locallyCompactSpace
  have := ChartedSpace.locallyCompactSpace H M
  have := I.secondCountableTopology
  have := ChartedSpace.secondCountable_of_sigmaCompact H M
  let _ : MetricSpace M := TopologicalSpace.metrizableSpaceMetric M
  -- it suffices to show that the integral of the function vanishes on any compact set `s`
  apply ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero' hf (fun s hs ↦ Eq.symm ?_)
  obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) := hs.exists_isCompact_cthickening
  -- choose a sequence of smooth functions `gₙ` equal to `1` on `s` and vanishing outside of the
  -- `uₙ`-neighborhood of `s`, where `uₙ` tends to zero. Then each integral `∫ gₙ f` vanishes,
  -- and by dominated convergence these integrals converge to `∫ x in s, f`.
  obtain ⟨u, -, u_pos, u_lim⟩ : ∃ u, StrictAnti u ∧ (∀ (n : ℕ), u n ∈ Ioo 0 δ)
    ∧ Tendsto u atTop (𝓝 0) := exists_seq_strictAnti_tendsto' δpos
  let v : ℕ → Set M := fun n ↦ thickening (u n) s
  obtain ⟨K, K_compact, vK⟩ : ∃ K, IsCompact K ∧ ∀ n, v n ⊆ K :=
    ⟨_, hδ, fun n ↦ thickening_subset_cthickening_of_le (u_pos n).2.le _⟩
  have : ∀ n, ∃ (g : M → ℝ), support g = v n ∧ CMDiff ∞ g ∧ Set.range g ⊆ Set.Icc 0 1
          ∧ ∀ x ∈ s, g x = 1 := by
    intro n
    rcases exists_contMDiff_support_eq_eq_one_iff I isOpen_thickening hs.isClosed
      (self_subset_thickening (u_pos n).1 s) with ⟨g, g_smooth, g_range, g_supp, hg⟩
    exact ⟨g, g_supp, g_smooth, g_range, fun x hx ↦ (hg x).1 hx⟩
  choose g g_supp g_diff g_range hg using this
  -- main fact: the integral of `∫ gₙ f` tends to `∫ x in s, f`.
  have L : Tendsto (fun n ↦ ∫ x, g n x • f x ∂μ) atTop (𝓝 (∫ x in s, f x ∂μ)) := by
    rw [← integral_indicator hs.measurableSet]
    let bound : M → ℝ := K.indicator (fun x ↦ ‖f x‖)
    have A : ∀ n, AEStronglyMeasurable (fun x ↦ g n x • f x) μ :=
      fun n ↦ (g_diff n).continuous.aestronglyMeasurable.smul hf.aestronglyMeasurable
    have B : Integrable bound μ := by
      rw [integrable_indicator_iff K_compact.measurableSet]
      exact (hf.integrableOn_isCompact K_compact).norm
    have C : ∀ n, ∀ᵐ x ∂μ, ‖g n x • f x‖ ≤ bound x := by
      intro n
      filter_upwards with x
      rw [norm_smul]
      refine le_indicator_apply (fun _ ↦ ?_) (fun hxK ↦ ?_)
      · have : ‖g n x‖ ≤ 1 := by
          have := g_range n (mem_range_self (f := g n) x)
          rw [Real.norm_of_nonneg this.1]
          exact this.2
        exact mul_le_of_le_one_left (norm_nonneg _) this
      · have : g n x = 0 := by rw [← notMem_support, g_supp]; contrapose hxK; exact vK n hxK
        simp [this]
    have D : ∀ᵐ x ∂μ, Tendsto (fun n => g n x • f x) atTop (𝓝 (s.indicator f x)) := by
      filter_upwards with x
      by_cases hxs : x ∈ s
      · have : ∀ n, g n x = 1 := fun n ↦ hg n x hxs
        simp [this, indicator_of_mem hxs f]
      · simp_rw [indicator_of_notMem hxs f]
        apply tendsto_const_nhds.congr'
        suffices H : ∀ᶠ n in atTop, g n x = 0 by
          filter_upwards [H] with n hn using by simp [hn]
        obtain ⟨ε, εpos, hε⟩ : ∃ ε, 0 < ε ∧ x ∉ thickening ε s := by
          rw [← hs.isClosed.closure_eq, closure_eq_iInter_thickening s] at hxs
          simpa using hxs
        filter_upwards [(tendsto_order.1 u_lim).2 _ εpos] with n hn
        rw [← notMem_support, g_supp]
        contrapose hε
        exact thickening_mono hn.le s hε
    exact tendsto_integral_of_dominated_convergence bound A B C D
  -- deduce that `∫ x in s, f = 0` as each integral `∫ gₙ f` vanishes by assumption
  have : ∀ n, ∫ x, g n x • f x ∂μ = 0 := by
    refine fun n ↦ h _ (g_diff n) ?_
    apply HasCompactSupport.of_support_subset_isCompact K_compact
    simpa [g_supp] using vK n
  simpa [this] using L

Goal state after: