l3lab/ntp-mathlib huggingface dataset:
/-- If a function `g` is locally integrable, then the convolution `φ i * g` converges almost
everywhere to `g` if `φ i` is a sequence of bump functions with support tending to `0`, provided
that the ratio between the inner and outer radii of `φ i` remains bounded. -/
theorem ae_convolution_tendsto_right_of_locallyIntegrable
    {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
    (hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
    (h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
    Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) := by
  have : IsAddHaarMeasure μ := ⟨⟩
  -- By Lebesgue differentiation theorem, the average of `g` on a small ball converges
  -- almost everywhere to the value of `g` as the radius shrinks to zero.
  -- We will see that this set of points satisfies the desired conclusion.
  filter_upwards [(Besicovitch.vitaliFamily μ).ae_tendsto_average_norm_sub hg] with x₀ h₀
  simp only [convolution_eq_swap, lsmul_apply]
  have hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0) :=
    tendsto_nhdsWithin_iff.2 ⟨hφ, eventually_of_forall (fun i ↦ (φ i).rOut_pos)⟩
  have := (h₀.comp (Besicovitch.tendsto_filterAt μ x₀)).comp hφ'
  simp only [Function.comp] at this
  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (FiniteDimensional.finrank ℝ G)) this
  · apply eventually_of_forall (fun i ↦ ?_)
    

Proof state:
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
φ✝ : ContDiffBump 0
x₀✝ : G
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : IsOpenPosMeasure μ
inst✝¹ : FiniteDimensional ℝ G
inst✝ : IsAddLeftInvariant μ
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g
this✝ : IsAddHaarMeasure μ
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (fun x => ⨍ (y : G) in closedBall x₀ (φ x).rOut, ‖g y - g x₀‖ ∂μ) l (𝓝 0)
i : ι
⊢ IntegrableOn (fun y => g y) (closedBall x₀ (φ i).rOut)

Next tactic:
apply hg.integrableOn_isCompact
/-- If a function `g` is locally integrable, then the convolution `φ i * g` converges almost
everywhere to `g` if `φ i` is a sequence of bump functions with support tending to `0`, provided
that the ratio between the inner and outer radii of `φ i` remains bounded. -/
theorem ae_convolution_tendsto_right_of_locallyIntegrable
    {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
    (hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
    (h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
    Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) := by
  have : IsAddHaarMeasure μ := ⟨⟩
  -- By Lebesgue differentiation theorem, the average of `g` on a small ball converges
  -- almost everywhere to the value of `g` as the radius shrinks to zero.
  -- We will see that this set of points satisfies the desired conclusion.
  filter_upwards [(Besicovitch.vitaliFamily μ).ae_tendsto_average_norm_sub hg] with x₀ h₀
  simp only [convolution_eq_swap, lsmul_apply]
  have hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0) :=
    tendsto_nhdsWithin_iff.2 ⟨hφ, eventually_of_forall (fun i ↦ (φ i).rOut_pos)⟩
  have := (h₀.comp (Besicovitch.tendsto_filterAt μ x₀)).comp hφ'
  simp only [Function.comp] at this
  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (FiniteDimensional.finrank ℝ G)) this
  · apply eventually_of_forall (fun i ↦ ?_)
    apply hg.integrableOn_isCompact
    

Proof state:
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
φ✝ : ContDiffBump 0
x₀✝ : G
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : IsOpenPosMeasure μ
inst✝¹ : FiniteDimensional ℝ G
inst✝ : IsAddLeftInvariant μ
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g
this✝ : IsAddHaarMeasure μ
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (fun x => ⨍ (y : G) in closedBall x₀ (φ x).rOut, ‖g y - g x₀‖ ∂μ) l (𝓝 0)
i : ι
⊢ IsCompact (closedBall x₀ (φ i).rOut)

Next tactic:
exact isCompact_closedBall _ _
/-- If a function `g` is locally integrable, then the convolution `φ i * g` converges almost
everywhere to `g` if `φ i` is a sequence of bump functions with support tending to `0`, provided
that the ratio between the inner and outer radii of `φ i` remains bounded. -/
theorem ae_convolution_tendsto_right_of_locallyIntegrable
    {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
    (hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
    (h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
    Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) := by
  have : IsAddHaarMeasure μ := ⟨⟩
  -- By Lebesgue differentiation theorem, the average of `g` on a small ball converges
  -- almost everywhere to the value of `g` as the radius shrinks to zero.
  -- We will see that this set of points satisfies the desired conclusion.
  filter_upwards [(Besicovitch.vitaliFamily μ).ae_tendsto_average_norm_sub hg] with x₀ h₀
  simp only [convolution_eq_swap, lsmul_apply]
  have hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0) :=
    tendsto_nhdsWithin_iff.2 ⟨hφ, eventually_of_forall (fun i ↦ (φ i).rOut_pos)⟩
  have := (h₀.comp (Besicovitch.tendsto_filterAt μ x₀)).comp hφ'
  simp only [Function.comp] at this
  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (FiniteDimensional.finrank ℝ G)) this
  · apply eventually_of_forall (fun i ↦ ?_)
    apply hg.integrableOn_isCompact
    exact isCompact_closedBall _ _
  · 

Proof state:
case h.hg
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
φ✝ : ContDiffBump 0
x₀✝ : G
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : IsOpenPosMeasure μ
inst✝¹ : FiniteDimensional ℝ G
inst✝ : IsAddLeftInvariant μ
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g
this✝ : IsAddHaarMeasure μ
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (fun x => ⨍ (y : G) in closedBall x₀ (φ x).rOut, ‖g y - g x₀‖ ∂μ) l (𝓝 0)
⊢ Tendsto (fun i => ∫ (y : G), ContDiffBump.normed (φ i) μ (x₀ - y) ∂μ) l (𝓝 1)

Next tactic:
apply tendsto_const_nhds.congr (fun i ↦ ?_)
/-- If a function `g` is locally integrable, then the convolution `φ i * g` converges almost
everywhere to `g` if `φ i` is a sequence of bump functions with support tending to `0`, provided
that the ratio between the inner and outer radii of `φ i` remains bounded. -/
theorem ae_convolution_tendsto_right_of_locallyIntegrable
    {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
    (hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
    (h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
    Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) := by
  have : IsAddHaarMeasure μ := ⟨⟩
  -- By Lebesgue differentiation theorem, the average of `g` on a small ball converges
  -- almost everywhere to the value of `g` as the radius shrinks to zero.
  -- We will see that this set of points satisfies the desired conclusion.
  filter_upwards [(Besicovitch.vitaliFamily μ).ae_tendsto_average_norm_sub hg] with x₀ h₀
  simp only [convolution_eq_swap, lsmul_apply]
  have hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0) :=
    tendsto_nhdsWithin_iff.2 ⟨hφ, eventually_of_forall (fun i ↦ (φ i).rOut_pos)⟩
  have := (h₀.comp (Besicovitch.tendsto_filterAt μ x₀)).comp hφ'
  simp only [Function.comp] at this
  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (FiniteDimensional.finrank ℝ G)) this
  · apply eventually_of_forall (fun i ↦ ?_)
    apply hg.integrableOn_isCompact
    exact isCompact_closedBall _ _
  · apply tendsto_const_nhds.congr (fun i ↦ ?_)
    

Proof state:
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
φ✝ : ContDiffBump 0
x₀✝ : G
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : IsOpenPosMeasure μ
inst✝¹ : FiniteDimensional ℝ G
inst✝ : IsAddLeftInvariant μ
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g
this✝ : IsAddHaarMeasure μ
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (fun x => ⨍ (y : G) in closedBall x₀ (φ x).rOut, ‖g y - g x₀‖ ∂μ) l (𝓝 0)
i : ι
⊢ 1 = ∫ (y : G), ContDiffBump.normed (φ i) μ (x₀ - y) ∂μ

Next tactic:
rw [← integral_neg_eq_self]
/-- If a function `g` is locally integrable, then the convolution `φ i * g` converges almost
everywhere to `g` if `φ i` is a sequence of bump functions with support tending to `0`, provided
that the ratio between the inner and outer radii of `φ i` remains bounded. -/
theorem ae_convolution_tendsto_right_of_locallyIntegrable
    {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
    (hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
    (h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
    Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) := by
  have : IsAddHaarMeasure μ := ⟨⟩
  -- By Lebesgue differentiation theorem, the average of `g` on a small ball converges
  -- almost everywhere to the value of `g` as the radius shrinks to zero.
  -- We will see that this set of points satisfies the desired conclusion.
  filter_upwards [(Besicovitch.vitaliFamily μ).ae_tendsto_average_norm_sub hg] with x₀ h₀
  simp only [convolution_eq_swap, lsmul_apply]
  have hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0) :=
    tendsto_nhdsWithin_iff.2 ⟨hφ, eventually_of_forall (fun i ↦ (φ i).rOut_pos)⟩
  have := (h₀.comp (Besicovitch.tendsto_filterAt μ x₀)).comp hφ'
  simp only [Function.comp] at this
  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (FiniteDimensional.finrank ℝ G)) this
  · apply eventually_of_forall (fun i ↦ ?_)
    apply hg.integrableOn_isCompact
    exact isCompact_closedBall _ _
  · apply tendsto_const_nhds.congr (fun i ↦ ?_)
    rw [← integral_neg_eq_self]
    

Proof state:
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
φ✝ : ContDiffBump 0
x₀✝ : G
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : IsOpenPosMeasure μ
inst✝¹ : FiniteDimensional ℝ G
inst✝ : IsAddLeftInvariant μ
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g
this✝ : IsAddHaarMeasure μ
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (fun x => ⨍ (y : G) in closedBall x₀ (φ x).rOut, ‖g y - g x₀‖ ∂μ) l (𝓝 0)
i : ι
⊢ 1 = ∫ (x : G), ContDiffBump.normed (φ i) μ (x₀ - -x) ∂μ

Next tactic:
simp only [sub_neg_eq_add, integral_add_left_eq_self, integral_normed]
/-- If a function `g` is locally integrable, then the convolution `φ i * g` converges almost
everywhere to `g` if `φ i` is a sequence of bump functions with support tending to `0`, provided
that the ratio between the inner and outer radii of `φ i` remains bounded. -/
theorem ae_convolution_tendsto_right_of_locallyIntegrable
    {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
    (hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
    (h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
    Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) := by
  have : IsAddHaarMeasure μ := ⟨⟩
  -- By Lebesgue differentiation theorem, the average of `g` on a small ball converges
  -- almost everywhere to the value of `g` as the radius shrinks to zero.
  -- We will see that this set of points satisfies the desired conclusion.
  filter_upwards [(Besicovitch.vitaliFamily μ).ae_tendsto_average_norm_sub hg] with x₀ h₀
  simp only [convolution_eq_swap, lsmul_apply]
  have hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0) :=
    tendsto_nhdsWithin_iff.2 ⟨hφ, eventually_of_forall (fun i ↦ (φ i).rOut_pos)⟩
  have := (h₀.comp (Besicovitch.tendsto_filterAt μ x₀)).comp hφ'
  simp only [Function.comp] at this
  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (FiniteDimensional.finrank ℝ G)) this
  · apply eventually_of_forall (fun i ↦ ?_)
    apply hg.integrableOn_isCompact
    exact isCompact_closedBall _ _
  · apply tendsto_const_nhds.congr (fun i ↦ ?_)
    rw [← integral_neg_eq_self]
    simp only [sub_neg_eq_add, integral_add_left_eq_self, integral_normed]
  · 

Proof state:
case h.g_supp
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
φ✝ : ContDiffBump 0
x₀✝ : G
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : IsOpenPosMeasure μ
inst✝¹ : FiniteDimensional ℝ G
inst✝ : IsAddLeftInvariant μ
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g
this✝ : IsAddHaarMeasure μ
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (fun x => ⨍ (y : G) in closedBall x₀ (φ x).rOut, ‖g y - g x₀‖ ∂μ) l (𝓝 0)
⊢ ∀ᶠ (i : ι) in l, (support fun y => ContDiffBump.normed (φ i) μ (x₀ - y)) ⊆ closedBall x₀ (φ i).rOut

Next tactic:
apply eventually_of_forall (fun i ↦ ?_)
/-- If a function `g` is locally integrable, then the convolution `φ i * g` converges almost
everywhere to `g` if `φ i` is a sequence of bump functions with support tending to `0`, provided
that the ratio between the inner and outer radii of `φ i` remains bounded. -/
theorem ae_convolution_tendsto_right_of_locallyIntegrable
    {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
    (hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
    (h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
    Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) := by
  have : IsAddHaarMeasure μ := ⟨⟩
  -- By Lebesgue differentiation theorem, the average of `g` on a small ball converges
  -- almost everywhere to the value of `g` as the radius shrinks to zero.
  -- We will see that this set of points satisfies the desired conclusion.
  filter_upwards [(Besicovitch.vitaliFamily μ).ae_tendsto_average_norm_sub hg] with x₀ h₀
  simp only [convolution_eq_swap, lsmul_apply]
  have hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0) :=
    tendsto_nhdsWithin_iff.2 ⟨hφ, eventually_of_forall (fun i ↦ (φ i).rOut_pos)⟩
  have := (h₀.comp (Besicovitch.tendsto_filterAt μ x₀)).comp hφ'
  simp only [Function.comp] at this
  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (FiniteDimensional.finrank ℝ G)) this
  · apply eventually_of_forall (fun i ↦ ?_)
    apply hg.integrableOn_isCompact
    exact isCompact_closedBall _ _
  · apply tendsto_const_nhds.congr (fun i ↦ ?_)
    rw [← integral_neg_eq_self]
    simp only [sub_neg_eq_add, integral_add_left_eq_self, integral_normed]
  · apply eventually_of_forall (fun i ↦ ?_)
    

Proof state:
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
φ✝ : ContDiffBump 0
x₀✝ : G
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : IsOpenPosMeasure μ
inst✝¹ : FiniteDimensional ℝ G
inst✝ : IsAddLeftInvariant μ
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g
this✝ : IsAddHaarMeasure μ
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (fun x => ⨍ (y : G) in closedBall x₀ (φ x).rOut, ‖g y - g x₀‖ ∂μ) l (𝓝 0)
i : ι
⊢ (support fun y => ContDiffBump.normed (φ i) μ (x₀ - y)) ⊆ closedBall x₀ (φ i).rOut

Next tactic:
change support ((ContDiffBump.normed (φ i) μ) ∘ (fun y ↦ x₀ - y)) ⊆ closedBall x₀ (φ i).rOut
/-- If a function `g` is locally integrable, then the convolution `φ i * g` converges almost
everywhere to `g` if `φ i` is a sequence of bump functions with support tending to `0`, provided
that the ratio between the inner and outer radii of `φ i` remains bounded. -/
theorem ae_convolution_tendsto_right_of_locallyIntegrable
    {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
    (hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
    (h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
    Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) := by
  have : IsAddHaarMeasure μ := ⟨⟩
  -- By Lebesgue differentiation theorem, the average of `g` on a small ball converges
  -- almost everywhere to the value of `g` as the radius shrinks to zero.
  -- We will see that this set of points satisfies the desired conclusion.
  filter_upwards [(Besicovitch.vitaliFamily μ).ae_tendsto_average_norm_sub hg] with x₀ h₀
  simp only [convolution_eq_swap, lsmul_apply]
  have hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0) :=
    tendsto_nhdsWithin_iff.2 ⟨hφ, eventually_of_forall (fun i ↦ (φ i).rOut_pos)⟩
  have := (h₀.comp (Besicovitch.tendsto_filterAt μ x₀)).comp hφ'
  simp only [Function.comp] at this
  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (FiniteDimensional.finrank ℝ G)) this
  · apply eventually_of_forall (fun i ↦ ?_)
    apply hg.integrableOn_isCompact
    exact isCompact_closedBall _ _
  · apply tendsto_const_nhds.congr (fun i ↦ ?_)
    rw [← integral_neg_eq_self]
    simp only [sub_neg_eq_add, integral_add_left_eq_self, integral_normed]
  · apply eventually_of_forall (fun i ↦ ?_)
    change support ((ContDiffBump.normed (φ i) μ) ∘ (fun y ↦ x₀ - y)) ⊆ closedBall x₀ (φ i).rOut
    

Proof state:
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
φ✝ : ContDiffBump 0
x₀✝ : G
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : IsOpenPosMeasure μ
inst✝¹ : FiniteDimensional ℝ G
inst✝ : IsAddLeftInvariant μ
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g
this✝ : IsAddHaarMeasure μ
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (fun x => ⨍ (y : G) in closedBall x₀ (φ x).rOut, ‖g y - g x₀‖ ∂μ) l (𝓝 0)
i : ι
⊢ support (ContDiffBump.normed (φ i) μ ∘ fun y => x₀ - y) ⊆ closedBall x₀ (φ i).rOut

Next tactic:
simp only [support_comp_eq_preimage, support_normed_eq]
/-- If a function `g` is locally integrable, then the convolution `φ i * g` converges almost
everywhere to `g` if `φ i` is a sequence of bump functions with support tending to `0`, provided
that the ratio between the inner and outer radii of `φ i` remains bounded. -/
theorem ae_convolution_tendsto_right_of_locallyIntegrable
    {ι} {φ : ι → ContDiffBump (0 : G)} {l : Filter ι} {K : ℝ}
    (hφ : Tendsto (fun i ↦ (φ i).rOut) l (𝓝 0))
    (h'φ : ∀ᶠ i in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : LocallyIntegrable g μ) : ∀ᵐ x₀ ∂μ,
    Tendsto (fun i ↦ ((φ i).normed μ ⋆[lsmul ℝ ℝ, μ] g) x₀) l (𝓝 (g x₀)) := by
  have : IsAddHaarMeasure μ := ⟨⟩
  -- By Lebesgue differentiation theorem, the average of `g` on a small ball converges
  -- almost everywhere to the value of `g` as the radius shrinks to zero.
  -- We will see that this set of points satisfies the desired conclusion.
  filter_upwards [(Besicovitch.vitaliFamily μ).ae_tendsto_average_norm_sub hg] with x₀ h₀
  simp only [convolution_eq_swap, lsmul_apply]
  have hφ' : Tendsto (fun i ↦ (φ i).rOut) l (𝓝[>] 0) :=
    tendsto_nhdsWithin_iff.2 ⟨hφ, eventually_of_forall (fun i ↦ (φ i).rOut_pos)⟩
  have := (h₀.comp (Besicovitch.tendsto_filterAt μ x₀)).comp hφ'
  simp only [Function.comp] at this
  apply tendsto_integral_smul_of_tendsto_average_norm_sub (K ^ (FiniteDimensional.finrank ℝ G)) this
  · apply eventually_of_forall (fun i ↦ ?_)
    apply hg.integrableOn_isCompact
    exact isCompact_closedBall _ _
  · apply tendsto_const_nhds.congr (fun i ↦ ?_)
    rw [← integral_neg_eq_self]
    simp only [sub_neg_eq_add, integral_add_left_eq_self, integral_normed]
  · apply eventually_of_forall (fun i ↦ ?_)
    change support ((ContDiffBump.normed (φ i) μ) ∘ (fun y ↦ x₀ - y)) ⊆ closedBall x₀ (φ i).rOut
    simp only [support_comp_eq_preimage, support_normed_eq]
    

Proof state:
G : Type uG
E' : Type uE'
inst✝¹¹ : NormedAddCommGroup E'
g : G → E'
inst✝¹⁰ : MeasurableSpace G
μ : Measure G
inst✝⁹ : NormedSpace ℝ E'
inst✝⁸ : NormedAddCommGroup G
inst✝⁷ : NormedSpace ℝ G
inst✝⁶ : HasContDiffBump G
inst✝⁵ : CompleteSpace E'
φ✝ : ContDiffBump 0
x₀✝ : G
inst✝⁴ : BorelSpace G
inst✝³ : IsLocallyFiniteMeasure μ
inst✝² : IsOpenPosMeasure μ
inst✝¹ : FiniteDimensional ℝ G
inst✝ : IsAddLeftInvariant μ
ι : Type u_1
φ : ι → ContDiffBump 0
l : Filter ι
K : ℝ
hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)
h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn
hg : LocallyIntegrable g
this✝ : IsAddHaarMeasure μ
x₀ : G
h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x₀) (𝓝 0)
hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)
this : Tendsto (fun x => ⨍ (y : G) in closedBall x₀ (φ x).rOut, ‖g y - g x₀‖ ∂μ) l (𝓝 0)
i : ι
⊢ (fun y => x₀ - y) ⁻¹' ball 0 (φ i).rOut ⊆ closedBall x₀ (φ i).rOut

Next tactic:
intro x hx