/-- If a locally integrable function `f` on a finite-dimensional real manifold has zero integral
when multiplied by any smooth compactly supported function, then `f` vanishes almost everywhere. -/
theorem ae_eq_zero_of_integral_smooth_smul_eq_zero (hf : LocallyIntegrable f μ)
    (h : ∀ g : M → ℝ, Smooth I 𝓘(ℝ) g → HasCompactSupport g → ∫ x, g x • f x ∂μ = 0) :
    ∀ᵐ x ∂μ, f x = 0 := by
  -- record topological properties of `M`
  have := I.locallyCompactSpace
  have := ChartedSpace.locallyCompactSpace H M
  have := I.secondCountableTopology
  have := ChartedSpace.secondCountable_of_sigma_compact H M
  have := ManifoldWithCorners.metrizableSpace I 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_set_integral_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 ∧ Smooth I 𝓘(ℝ) g ∧ Set.range g ⊆ Set.Icc 0 1
          ∧ ∀ x ∈ s, g x = 1 := by
    intro n
    rcases exists_msmooth_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⟩
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro
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 : SmoothManifoldWithCorners I M
inst✝³ : MeasurableSpace M
inst✝² : BorelSpace M
inst✝¹ : SigmaCompactSpace M
inst✝ : T2Space M
f f' : M → F
μ : Measure M
hf : LocallyIntegrable f
h : ∀ (g : M → ℝ), Smooth I 𝓘(ℝ, ℝ) g → HasCompactSupport g → ∫ (x : M), g x • f x ∂μ = 0
this✝⁴ : LocallyCompactSpace H
this✝³ : LocallyCompactSpace M
this✝² : SecondCountableTopology H
this✝¹ : SecondCountableTopology M
this✝ : MetrizableSpace M
x✝ : MetricSpace M := metrizableSpaceMetric M
s : Set M
hs : IsCompact s
δ : ℝ
δpos : 0 < δ
hδ : IsCompact (cthickening δ s)
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), u n ∈ Ioo 0 δ
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → Set M := fun n => thickening (u n) s
K : Set M
K_compact : IsCompact K
vK : ∀ (n : ℕ), v n ⊆ K
this : ∀ (n : ℕ), ∃ g, support g = v n ∧ Smooth I 𝓘(ℝ, ℝ) g ∧ range g ⊆ Icc 0 1 ∧ ∀ x ∈ s, g x = 1
⊢ 0 = ∫ (x : M) in s, f x ∂μ

Next tactic:
choose g g_supp g_diff g_range hg using this