l3lab/ntp-mathlib huggingface dataset:
/-- In a locally compact regular space with an inner regular measure, the measure of a compact
set `k` is the infimum of the integrals of compactly supported functions equal to `1` on `k`. -/
lemma IsCompact.measure_eq_biInf_integral_hasCompactSupport
    {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
    {k : Set X} (hk : IsCompact k)
    (μ : Measure X) [IsFiniteMeasureOnCompacts μ] [InnerRegularCompactLTTop μ]
    [LocallyCompactSpace X] [RegularSpace X] :
    μ k = ⨅ (f : X → ℝ) (_ : Continuous f) (_ : HasCompactSupport f) (_ : EqOn f 1 k)
      (_ : 0 ≤ f), ENNReal.ofReal (∫ x, f x ∂μ) := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    intro f f_cont f_comp fk f_nonneg
    apply (f_cont.integrable_of_hasCompactSupport f_comp).measure_le_integral
    · exact eventually_of_forall f_nonneg
    · exact fun x hx ↦ by simp [fk hx]
  · apply le_of_forall_lt' (fun r hr ↦ ?_)
    simp only [iInf_lt_iff, exists_prop, exists_and_left]
    obtain ⟨U, kU, U_open, mu_U⟩ : ∃ U, k ⊆ U ∧ IsOpen U ∧ μ U < r :=
      hk.exists_isOpen_lt_of_lt r hr
    

Proof state:
case intro.intro.intro
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
k : Set X
hk : IsCompact k
μ : Measure X
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : InnerRegularCompactLTTop μ
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
r : ℝ≥0∞
hr : ↑↑μ k < r
U : Set X
kU : k ⊆ U
U_open : IsOpen U
mu_U : ↑↑μ U < r
⊢ ∃ i, Continuous i ∧ HasCompactSupport i ∧ EqOn i 1 k ∧ 0 ≤ i ∧ ENNReal.ofReal (∫ (x : X), i x ∂μ) < r

Next tactic:
obtain ⟨⟨f, f_cont⟩, fk, fU, f_comp, f_range⟩ : ∃ (f : C(X, ℝ)), EqOn f 1 k ∧ EqOn f 0 Uᶜ
        ∧ HasCompactSupport f ∧ ∀ (x : X), f x ∈ Icc 0 1 := exists_continuous_one_zero_of_isCompact
      hk U_open.isClosed_compl (disjoint_compl_right_iff_subset.mpr kU)
/-- In a locally compact regular space with an inner regular measure, the measure of a compact
set `k` is the infimum of the integrals of compactly supported functions equal to `1` on `k`. -/
lemma IsCompact.measure_eq_biInf_integral_hasCompactSupport
    {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
    {k : Set X} (hk : IsCompact k)
    (μ : Measure X) [IsFiniteMeasureOnCompacts μ] [InnerRegularCompactLTTop μ]
    [LocallyCompactSpace X] [RegularSpace X] :
    μ k = ⨅ (f : X → ℝ) (_ : Continuous f) (_ : HasCompactSupport f) (_ : EqOn f 1 k)
      (_ : 0 ≤ f), ENNReal.ofReal (∫ x, f x ∂μ) := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    intro f f_cont f_comp fk f_nonneg
    apply (f_cont.integrable_of_hasCompactSupport f_comp).measure_le_integral
    · exact eventually_of_forall f_nonneg
    · exact fun x hx ↦ by simp [fk hx]
  · apply le_of_forall_lt' (fun r hr ↦ ?_)
    simp only [iInf_lt_iff, exists_prop, exists_and_left]
    obtain ⟨U, kU, U_open, mu_U⟩ : ∃ U, k ⊆ U ∧ IsOpen U ∧ μ U < r :=
      hk.exists_isOpen_lt_of_lt r hr
    obtain ⟨⟨f, f_cont⟩, fk, fU, f_comp, f_range⟩ : ∃ (f : C(X, ℝ)), EqOn f 1 k ∧ EqOn f 0 Uᶜ
        ∧ HasCompactSupport f ∧ ∀ (x : X), f x ∈ Icc 0 1 := exists_continuous_one_zero_of_isCompact
      hk U_open.isClosed_compl (disjoint_compl_right_iff_subset.mpr kU)
    

Proof state:
case intro.intro.intro.intro.mk.intro.intro.intro
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
k : Set X
hk : IsCompact k
μ : Measure X
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : InnerRegularCompactLTTop μ
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
r : ℝ≥0∞
hr : ↑↑μ k < r
U : Set X
kU : k ⊆ U
U_open : IsOpen U
mu_U : ↑↑μ U < r
f : X → ℝ
f_cont : Continuous f
fk : EqOn (⇑(ContinuousMap.mk f)) 1 k
fU : EqOn (⇑(ContinuousMap.mk f)) 0 Uᶜ
f_comp : HasCompactSupport ⇑(ContinuousMap.mk f)
f_range : ∀ (x : X), (ContinuousMap.mk f) x ∈ Icc 0 1
⊢ ∃ i, Continuous i ∧ HasCompactSupport i ∧ EqOn i 1 k ∧ 0 ≤ i ∧ ENNReal.ofReal (∫ (x : X), i x ∂μ) < r

Next tactic:
refine ⟨f, f_cont, f_comp, fk, fun x ↦ (f_range x).1, ?_⟩
/-- In a locally compact regular space with an inner regular measure, the measure of a compact
set `k` is the infimum of the integrals of compactly supported functions equal to `1` on `k`. -/
lemma IsCompact.measure_eq_biInf_integral_hasCompactSupport
    {X : Type*} [TopologicalSpace X] [MeasurableSpace X] [BorelSpace X]
    {k : Set X} (hk : IsCompact k)
    (μ : Measure X) [IsFiniteMeasureOnCompacts μ] [InnerRegularCompactLTTop μ]
    [LocallyCompactSpace X] [RegularSpace X] :
    μ k = ⨅ (f : X → ℝ) (_ : Continuous f) (_ : HasCompactSupport f) (_ : EqOn f 1 k)
      (_ : 0 ≤ f), ENNReal.ofReal (∫ x, f x ∂μ) := by
  apply le_antisymm
  · simp only [le_iInf_iff]
    intro f f_cont f_comp fk f_nonneg
    apply (f_cont.integrable_of_hasCompactSupport f_comp).measure_le_integral
    · exact eventually_of_forall f_nonneg
    · exact fun x hx ↦ by simp [fk hx]
  · apply le_of_forall_lt' (fun r hr ↦ ?_)
    simp only [iInf_lt_iff, exists_prop, exists_and_left]
    obtain ⟨U, kU, U_open, mu_U⟩ : ∃ U, k ⊆ U ∧ IsOpen U ∧ μ U < r :=
      hk.exists_isOpen_lt_of_lt r hr
    obtain ⟨⟨f, f_cont⟩, fk, fU, f_comp, f_range⟩ : ∃ (f : C(X, ℝ)), EqOn f 1 k ∧ EqOn f 0 Uᶜ
        ∧ HasCompactSupport f ∧ ∀ (x : X), f x ∈ Icc 0 1 := exists_continuous_one_zero_of_isCompact
      hk U_open.isClosed_compl (disjoint_compl_right_iff_subset.mpr kU)
    refine ⟨f, f_cont, f_comp, fk, fun x ↦ (f_range x).1, ?_⟩
    

Proof state:
case intro.intro.intro.intro.mk.intro.intro.intro
X : Type u_1
inst✝⁶ : TopologicalSpace X
inst✝⁵ : MeasurableSpace X
inst✝⁴ : BorelSpace X
k : Set X
hk : IsCompact k
μ : Measure X
inst✝³ : IsFiniteMeasureOnCompacts μ
inst✝² : InnerRegularCompactLTTop μ
inst✝¹ : LocallyCompactSpace X
inst✝ : RegularSpace X
r : ℝ≥0∞
hr : ↑↑μ k < r
U : Set X
kU : k ⊆ U
U_open : IsOpen U
mu_U : ↑↑μ U < r
f : X → ℝ
f_cont : Continuous f
fk : EqOn (⇑(ContinuousMap.mk f)) 1 k
fU : EqOn (⇑(ContinuousMap.mk f)) 0 Uᶜ
f_comp : HasCompactSupport ⇑(ContinuousMap.mk f)
f_range : ∀ (x : X), (ContinuousMap.mk f) x ∈ Icc 0 1
⊢ ENNReal.ofReal (∫ (x : X), f x ∂μ) < r

Next tactic:
exact (integral_le_measure (fun x _hx ↦ (f_range x).2) (fun x hx ↦ (fU hx).le)).trans_lt mu_U
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  

Proof state:
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
⊢ Continuous fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ

Next tactic:
let k := tsupport g
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  

Proof state:
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
⊢ Continuous fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ

Next tactic:
have k_comp : IsCompact k := h'g
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  have k_comp : IsCompact k := h'g
  

Proof state:
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
k_comp : IsCompact k
⊢ Continuous fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ

Next tactic:
apply continuous_iff_continuousAt.2 (fun x₀ ↦ ?_)
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  have k_comp : IsCompact k := h'g
  apply continuous_iff_continuousAt.2 (fun x₀ ↦ ?_)
  

Proof state:
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
k_comp : IsCompact k
x₀ : G
⊢ ContinuousAt (fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ) x₀

Next tactic:
obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  have k_comp : IsCompact k := h'g
  apply continuous_iff_continuousAt.2 (fun x₀ ↦ ?_)
  obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
  

Proof state:
case intro.intro
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
⊢ ContinuousAt (fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ) x₀

Next tactic:
let k' : Set G := t • k⁻¹
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  have k_comp : IsCompact k := h'g
  apply continuous_iff_continuousAt.2 (fun x₀ ↦ ?_)
  obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
  let k' : Set G := t • k⁻¹
  

Proof state:
case intro.intro
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := t • k⁻¹
⊢ ContinuousAt (fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ) x₀

Next tactic:
have k'_comp : IsCompact k' := t_comp.smul_set k_comp.inv
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  have k_comp : IsCompact k := h'g
  apply continuous_iff_continuousAt.2 (fun x₀ ↦ ?_)
  obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
  let k' : Set G := t • k⁻¹
  have k'_comp : IsCompact k' := t_comp.smul_set k_comp.inv
  

Proof state:
case intro.intro
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := t • k⁻¹
k'_comp : IsCompact k'
⊢ ContinuousAt (fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ) x₀

Next tactic:
have A : ContinuousOn (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) t := by
    apply continuousOn_integral_of_compact_support k'_comp
    · exact (hg.comp (continuous_snd.inv.mul continuous_fst)).continuousOn
    · intro p x hp hx
      contrapose! hx
      refine ⟨p, p⁻¹ * x, hp, ?_, by simp⟩
      simpa only [Set.mem_inv, mul_inv_rev, inv_inv] using subset_tsupport _ hx
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  have k_comp : IsCompact k := h'g
  apply continuous_iff_continuousAt.2 (fun x₀ ↦ ?_)
  obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
  let k' : Set G := t • k⁻¹
  have k'_comp : IsCompact k' := t_comp.smul_set k_comp.inv
  have A : ContinuousOn (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) t := by
    

Proof state:
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := t • k⁻¹
k'_comp : IsCompact k'
⊢ ContinuousOn (fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ) t

Next tactic:
apply continuousOn_integral_of_compact_support k'_comp
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  have k_comp : IsCompact k := h'g
  apply continuous_iff_continuousAt.2 (fun x₀ ↦ ?_)
  obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
  let k' : Set G := t • k⁻¹
  have k'_comp : IsCompact k' := t_comp.smul_set k_comp.inv
  have A : ContinuousOn (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) t := by
    apply continuousOn_integral_of_compact_support k'_comp
    · 

Proof state:
case hf
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := t • k⁻¹
k'_comp : IsCompact k'
⊢ ContinuousOn (uncurry fun a x => g (x⁻¹ * a)) (t ×ˢ univ)

Next tactic:
exact (hg.comp (continuous_snd.inv.mul continuous_fst)).continuousOn
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  have k_comp : IsCompact k := h'g
  apply continuous_iff_continuousAt.2 (fun x₀ ↦ ?_)
  obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
  let k' : Set G := t • k⁻¹
  have k'_comp : IsCompact k' := t_comp.smul_set k_comp.inv
  have A : ContinuousOn (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) t := by
    apply continuousOn_integral_of_compact_support k'_comp
    · exact (hg.comp (continuous_snd.inv.mul continuous_fst)).continuousOn
    · 

Proof state:
case hfs
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := t • k⁻¹
k'_comp : IsCompact k'
⊢ ∀ (p x : G), p ∈ t → x ∉ k' → g (x⁻¹ * p) = 0

Next tactic:
intro p x hp hx
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  have k_comp : IsCompact k := h'g
  apply continuous_iff_continuousAt.2 (fun x₀ ↦ ?_)
  obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
  let k' : Set G := t • k⁻¹
  have k'_comp : IsCompact k' := t_comp.smul_set k_comp.inv
  have A : ContinuousOn (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) t := by
    apply continuousOn_integral_of_compact_support k'_comp
    · exact (hg.comp (continuous_snd.inv.mul continuous_fst)).continuousOn
    · intro p x hp hx
      

Proof state:
case hfs
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := t • k⁻¹
k'_comp : IsCompact k'
p x : G
hp : p ∈ t
hx : x ∉ k'
⊢ g (x⁻¹ * p) = 0

Next tactic:
contrapose! hx
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
    {G : Type*} [TopologicalSpace G] [LocallyCompactSpace G] [Group G] [TopologicalGroup G]
    [MeasurableSpace G] [BorelSpace G]
    {μ : Measure G} [IsFiniteMeasureOnCompacts μ] {E : Type*} [NormedAddCommGroup E]
    [NormedSpace ℝ E] {g : G → E}
    (hg : Continuous g) (h'g : HasCompactSupport g) :
    Continuous (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) := by
  let k := tsupport g
  have k_comp : IsCompact k := h'g
  apply continuous_iff_continuousAt.2 (fun x₀ ↦ ?_)
  obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
  let k' : Set G := t • k⁻¹
  have k'_comp : IsCompact k' := t_comp.smul_set k_comp.inv
  have A : ContinuousOn (fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂μ) t := by
    apply continuousOn_integral_of_compact_support k'_comp
    · exact (hg.comp (continuous_snd.inv.mul continuous_fst)).continuousOn
    · intro p x hp hx
      contrapose! hx
      

Proof state:
case hfs
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous g
h'g : HasCompactSupport g
k : Set G := tsupport g
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := t • k⁻¹
k'_comp : IsCompact k'
p x : G
hp : p ∈ t
hx : g (x⁻¹ * p) ≠ 0
⊢ x ∈ k'

Next tactic:
refine ⟨p, p⁻¹ * x, hp, ?_, by simp⟩