l3lab/ntp-mathlib huggingface dataset:
/-- In a group with a left invariant measure `μ` and a right invariant measure `ν`, one can express
integrals with respect to `μ` as integrals with respect to `ν` up to a constant scaling factor
(given in the statement as `∫ x, g x ∂μ` where `g` is a fixed reference function) and an
explicit density `y ↦ 1/∫ z, g (z⁻¹ * y) ∂ν`. -/
@[to_additive]
lemma integral_isMulLeftInvariant_isMulRightInvariant_combo
    {μ ν : Measure G} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν]
    [IsMulLeftInvariant μ] [IsMulRightInvariant ν] [IsOpenPosMeasure ν]
    {f g : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
    (hg : Continuous g) (h'g : HasCompactSupport g) (g_nonneg : 0 ≤ g) {x₀ : G} (g_pos : g x₀ ≠ 0) :
    ∫ x, f x ∂μ = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ := by
  let D : G → ℝ := fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂ν
  have D_cont : Continuous D := continuous_integral_apply_inv_mul hg h'g
  have D_pos : ∀ x, 0 < D x := by
    intro x
    have C : Continuous (fun y ↦ g (y⁻¹ * x)) := hg.comp (continuous_inv.mul continuous_const)
    apply (integral_pos_iff_support_of_nonneg _ _).2
    · apply C.isOpen_support.measure_pos ν
      exact ⟨x * x₀⁻¹, by simpa using g_pos⟩
    · exact fun y ↦ g_nonneg (y⁻¹ * x)
    · apply C.integrable_of_hasCompactSupport
      exact h'g.comp_homeomorph ((Homeomorph.inv G).trans (Homeomorph.mulRight x))
  calc
  ∫ x, f x ∂μ = ∫ x, f x * (D x)⁻¹ * D x ∂μ := by
    congr with x; rw [mul_assoc, inv_mul_cancel (D_pos x).ne', mul_one]
  _ = ∫ x, (∫ y, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂ν) ∂μ := by simp_rw [integral_mul_left]
  _ = ∫ y, (∫ x, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂μ) ∂ν := by
      apply integral_integral_swap_of_hasCompactSupport
      · apply Continuous.mul
        · exact (hf.comp continuous_fst).mul
            ((D_cont.comp continuous_fst).inv₀ (fun x ↦ (D_pos _).ne'))
        · exact hg.comp (continuous_snd.inv.mul continuous_fst)
      · let K := tsupport f
        

Proof state:
case h'f
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulRightInvariant ν
inst✝ : IsOpenPosMeasure ν
f g : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
hg : Continuous g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
D : G → ℝ := fun x => ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous D
D_pos : ∀ (x : G), 0 < D x
K : Set G := tsupport f
⊢ HasCompactSupport (uncurry fun x y => f x * (D x)⁻¹ * g (y⁻¹ * x))

Next tactic:
have K_comp : IsCompact K := h'f
/-- In a group with a left invariant measure `μ` and a right invariant measure `ν`, one can express
integrals with respect to `μ` as integrals with respect to `ν` up to a constant scaling factor
(given in the statement as `∫ x, g x ∂μ` where `g` is a fixed reference function) and an
explicit density `y ↦ 1/∫ z, g (z⁻¹ * y) ∂ν`. -/
@[to_additive]
lemma integral_isMulLeftInvariant_isMulRightInvariant_combo
    {μ ν : Measure G} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν]
    [IsMulLeftInvariant μ] [IsMulRightInvariant ν] [IsOpenPosMeasure ν]
    {f g : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
    (hg : Continuous g) (h'g : HasCompactSupport g) (g_nonneg : 0 ≤ g) {x₀ : G} (g_pos : g x₀ ≠ 0) :
    ∫ x, f x ∂μ = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ := by
  let D : G → ℝ := fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂ν
  have D_cont : Continuous D := continuous_integral_apply_inv_mul hg h'g
  have D_pos : ∀ x, 0 < D x := by
    intro x
    have C : Continuous (fun y ↦ g (y⁻¹ * x)) := hg.comp (continuous_inv.mul continuous_const)
    apply (integral_pos_iff_support_of_nonneg _ _).2
    · apply C.isOpen_support.measure_pos ν
      exact ⟨x * x₀⁻¹, by simpa using g_pos⟩
    · exact fun y ↦ g_nonneg (y⁻¹ * x)
    · apply C.integrable_of_hasCompactSupport
      exact h'g.comp_homeomorph ((Homeomorph.inv G).trans (Homeomorph.mulRight x))
  calc
  ∫ x, f x ∂μ = ∫ x, f x * (D x)⁻¹ * D x ∂μ := by
    congr with x; rw [mul_assoc, inv_mul_cancel (D_pos x).ne', mul_one]
  _ = ∫ x, (∫ y, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂ν) ∂μ := by simp_rw [integral_mul_left]
  _ = ∫ y, (∫ x, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂μ) ∂ν := by
      apply integral_integral_swap_of_hasCompactSupport
      · apply Continuous.mul
        · exact (hf.comp continuous_fst).mul
            ((D_cont.comp continuous_fst).inv₀ (fun x ↦ (D_pos _).ne'))
        · exact hg.comp (continuous_snd.inv.mul continuous_fst)
      · let K := tsupport f
        have K_comp : IsCompact K := h'f
        

Proof state:
case h'f
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulRightInvariant ν
inst✝ : IsOpenPosMeasure ν
f g : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
hg : Continuous g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
D : G → ℝ := fun x => ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous D
D_pos : ∀ (x : G), 0 < D x
K : Set G := tsupport f
K_comp : IsCompact K
⊢ HasCompactSupport (uncurry fun x y => f x * (D x)⁻¹ * g (y⁻¹ * x))

Next tactic:
let L := tsupport g
/-- In a group with a left invariant measure `μ` and a right invariant measure `ν`, one can express
integrals with respect to `μ` as integrals with respect to `ν` up to a constant scaling factor
(given in the statement as `∫ x, g x ∂μ` where `g` is a fixed reference function) and an
explicit density `y ↦ 1/∫ z, g (z⁻¹ * y) ∂ν`. -/
@[to_additive]
lemma integral_isMulLeftInvariant_isMulRightInvariant_combo
    {μ ν : Measure G} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν]
    [IsMulLeftInvariant μ] [IsMulRightInvariant ν] [IsOpenPosMeasure ν]
    {f g : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
    (hg : Continuous g) (h'g : HasCompactSupport g) (g_nonneg : 0 ≤ g) {x₀ : G} (g_pos : g x₀ ≠ 0) :
    ∫ x, f x ∂μ = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ := by
  let D : G → ℝ := fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂ν
  have D_cont : Continuous D := continuous_integral_apply_inv_mul hg h'g
  have D_pos : ∀ x, 0 < D x := by
    intro x
    have C : Continuous (fun y ↦ g (y⁻¹ * x)) := hg.comp (continuous_inv.mul continuous_const)
    apply (integral_pos_iff_support_of_nonneg _ _).2
    · apply C.isOpen_support.measure_pos ν
      exact ⟨x * x₀⁻¹, by simpa using g_pos⟩
    · exact fun y ↦ g_nonneg (y⁻¹ * x)
    · apply C.integrable_of_hasCompactSupport
      exact h'g.comp_homeomorph ((Homeomorph.inv G).trans (Homeomorph.mulRight x))
  calc
  ∫ x, f x ∂μ = ∫ x, f x * (D x)⁻¹ * D x ∂μ := by
    congr with x; rw [mul_assoc, inv_mul_cancel (D_pos x).ne', mul_one]
  _ = ∫ x, (∫ y, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂ν) ∂μ := by simp_rw [integral_mul_left]
  _ = ∫ y, (∫ x, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂μ) ∂ν := by
      apply integral_integral_swap_of_hasCompactSupport
      · apply Continuous.mul
        · exact (hf.comp continuous_fst).mul
            ((D_cont.comp continuous_fst).inv₀ (fun x ↦ (D_pos _).ne'))
        · exact hg.comp (continuous_snd.inv.mul continuous_fst)
      · let K := tsupport f
        have K_comp : IsCompact K := h'f
        let L := tsupport g
        

Proof state:
case h'f
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulRightInvariant ν
inst✝ : IsOpenPosMeasure ν
f g : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
hg : Continuous g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
D : G → ℝ := fun x => ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous D
D_pos : ∀ (x : G), 0 < D x
K : Set G := tsupport f
K_comp : IsCompact K
L : Set G := tsupport g
⊢ HasCompactSupport (uncurry fun x y => f x * (D x)⁻¹ * g (y⁻¹ * x))

Next tactic:
have L_comp : IsCompact L := h'g
/-- In a group with a left invariant measure `μ` and a right invariant measure `ν`, one can express
integrals with respect to `μ` as integrals with respect to `ν` up to a constant scaling factor
(given in the statement as `∫ x, g x ∂μ` where `g` is a fixed reference function) and an
explicit density `y ↦ 1/∫ z, g (z⁻¹ * y) ∂ν`. -/
@[to_additive]
lemma integral_isMulLeftInvariant_isMulRightInvariant_combo
    {μ ν : Measure G} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν]
    [IsMulLeftInvariant μ] [IsMulRightInvariant ν] [IsOpenPosMeasure ν]
    {f g : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
    (hg : Continuous g) (h'g : HasCompactSupport g) (g_nonneg : 0 ≤ g) {x₀ : G} (g_pos : g x₀ ≠ 0) :
    ∫ x, f x ∂μ = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ := by
  let D : G → ℝ := fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂ν
  have D_cont : Continuous D := continuous_integral_apply_inv_mul hg h'g
  have D_pos : ∀ x, 0 < D x := by
    intro x
    have C : Continuous (fun y ↦ g (y⁻¹ * x)) := hg.comp (continuous_inv.mul continuous_const)
    apply (integral_pos_iff_support_of_nonneg _ _).2
    · apply C.isOpen_support.measure_pos ν
      exact ⟨x * x₀⁻¹, by simpa using g_pos⟩
    · exact fun y ↦ g_nonneg (y⁻¹ * x)
    · apply C.integrable_of_hasCompactSupport
      exact h'g.comp_homeomorph ((Homeomorph.inv G).trans (Homeomorph.mulRight x))
  calc
  ∫ x, f x ∂μ = ∫ x, f x * (D x)⁻¹ * D x ∂μ := by
    congr with x; rw [mul_assoc, inv_mul_cancel (D_pos x).ne', mul_one]
  _ = ∫ x, (∫ y, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂ν) ∂μ := by simp_rw [integral_mul_left]
  _ = ∫ y, (∫ x, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂μ) ∂ν := by
      apply integral_integral_swap_of_hasCompactSupport
      · apply Continuous.mul
        · exact (hf.comp continuous_fst).mul
            ((D_cont.comp continuous_fst).inv₀ (fun x ↦ (D_pos _).ne'))
        · exact hg.comp (continuous_snd.inv.mul continuous_fst)
      · let K := tsupport f
        have K_comp : IsCompact K := h'f
        let L := tsupport g
        have L_comp : IsCompact L := h'g
        

Proof state:
case h'f
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulRightInvariant ν
inst✝ : IsOpenPosMeasure ν
f g : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
hg : Continuous g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
D : G → ℝ := fun x => ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous D
D_pos : ∀ (x : G), 0 < D x
K : Set G := tsupport f
K_comp : IsCompact K
L : Set G := tsupport g
L_comp : IsCompact L
⊢ HasCompactSupport (uncurry fun x y => f x * (D x)⁻¹ * g (y⁻¹ * x))

Next tactic:
let M := (fun (p : G × G) ↦ p.1 * p.2⁻¹) '' (K ×ˢ L)
/-- In a group with a left invariant measure `μ` and a right invariant measure `ν`, one can express
integrals with respect to `μ` as integrals with respect to `ν` up to a constant scaling factor
(given in the statement as `∫ x, g x ∂μ` where `g` is a fixed reference function) and an
explicit density `y ↦ 1/∫ z, g (z⁻¹ * y) ∂ν`. -/
@[to_additive]
lemma integral_isMulLeftInvariant_isMulRightInvariant_combo
    {μ ν : Measure G} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν]
    [IsMulLeftInvariant μ] [IsMulRightInvariant ν] [IsOpenPosMeasure ν]
    {f g : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
    (hg : Continuous g) (h'g : HasCompactSupport g) (g_nonneg : 0 ≤ g) {x₀ : G} (g_pos : g x₀ ≠ 0) :
    ∫ x, f x ∂μ = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ := by
  let D : G → ℝ := fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂ν
  have D_cont : Continuous D := continuous_integral_apply_inv_mul hg h'g
  have D_pos : ∀ x, 0 < D x := by
    intro x
    have C : Continuous (fun y ↦ g (y⁻¹ * x)) := hg.comp (continuous_inv.mul continuous_const)
    apply (integral_pos_iff_support_of_nonneg _ _).2
    · apply C.isOpen_support.measure_pos ν
      exact ⟨x * x₀⁻¹, by simpa using g_pos⟩
    · exact fun y ↦ g_nonneg (y⁻¹ * x)
    · apply C.integrable_of_hasCompactSupport
      exact h'g.comp_homeomorph ((Homeomorph.inv G).trans (Homeomorph.mulRight x))
  calc
  ∫ x, f x ∂μ = ∫ x, f x * (D x)⁻¹ * D x ∂μ := by
    congr with x; rw [mul_assoc, inv_mul_cancel (D_pos x).ne', mul_one]
  _ = ∫ x, (∫ y, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂ν) ∂μ := by simp_rw [integral_mul_left]
  _ = ∫ y, (∫ x, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂μ) ∂ν := by
      apply integral_integral_swap_of_hasCompactSupport
      · apply Continuous.mul
        · exact (hf.comp continuous_fst).mul
            ((D_cont.comp continuous_fst).inv₀ (fun x ↦ (D_pos _).ne'))
        · exact hg.comp (continuous_snd.inv.mul continuous_fst)
      · let K := tsupport f
        have K_comp : IsCompact K := h'f
        let L := tsupport g
        have L_comp : IsCompact L := h'g
        let M := (fun (p : G × G) ↦ p.1 * p.2⁻¹) '' (K ×ˢ L)
        

Proof state:
case h'f
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulRightInvariant ν
inst✝ : IsOpenPosMeasure ν
f g : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
hg : Continuous g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
D : G → ℝ := fun x => ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous D
D_pos : ∀ (x : G), 0 < D x
K : Set G := tsupport f
K_comp : IsCompact K
L : Set G := tsupport g
L_comp : IsCompact L
M : Set G := (fun p => p.1 * p.2⁻¹) '' K ×ˢ L
⊢ HasCompactSupport (uncurry fun x y => f x * (D x)⁻¹ * g (y⁻¹ * x))

Next tactic:
have M_comp : IsCompact M :=
          (K_comp.prod L_comp).image (continuous_fst.mul continuous_snd.inv)
/-- In a group with a left invariant measure `μ` and a right invariant measure `ν`, one can express
integrals with respect to `μ` as integrals with respect to `ν` up to a constant scaling factor
(given in the statement as `∫ x, g x ∂μ` where `g` is a fixed reference function) and an
explicit density `y ↦ 1/∫ z, g (z⁻¹ * y) ∂ν`. -/
@[to_additive]
lemma integral_isMulLeftInvariant_isMulRightInvariant_combo
    {μ ν : Measure G} [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts ν]
    [IsMulLeftInvariant μ] [IsMulRightInvariant ν] [IsOpenPosMeasure ν]
    {f g : G → ℝ} (hf : Continuous f) (h'f : HasCompactSupport f)
    (hg : Continuous g) (h'g : HasCompactSupport g) (g_nonneg : 0 ≤ g) {x₀ : G} (g_pos : g x₀ ≠ 0) :
    ∫ x, f x ∂μ = (∫ y, f y * (∫ z, g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ x, g x ∂μ := by
  let D : G → ℝ := fun (x : G) ↦ ∫ y, g (y⁻¹ * x) ∂ν
  have D_cont : Continuous D := continuous_integral_apply_inv_mul hg h'g
  have D_pos : ∀ x, 0 < D x := by
    intro x
    have C : Continuous (fun y ↦ g (y⁻¹ * x)) := hg.comp (continuous_inv.mul continuous_const)
    apply (integral_pos_iff_support_of_nonneg _ _).2
    · apply C.isOpen_support.measure_pos ν
      exact ⟨x * x₀⁻¹, by simpa using g_pos⟩
    · exact fun y ↦ g_nonneg (y⁻¹ * x)
    · apply C.integrable_of_hasCompactSupport
      exact h'g.comp_homeomorph ((Homeomorph.inv G).trans (Homeomorph.mulRight x))
  calc
  ∫ x, f x ∂μ = ∫ x, f x * (D x)⁻¹ * D x ∂μ := by
    congr with x; rw [mul_assoc, inv_mul_cancel (D_pos x).ne', mul_one]
  _ = ∫ x, (∫ y, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂ν) ∂μ := by simp_rw [integral_mul_left]
  _ = ∫ y, (∫ x, f x * (D x)⁻¹ * g (y⁻¹ * x) ∂μ) ∂ν := by
      apply integral_integral_swap_of_hasCompactSupport
      · apply Continuous.mul
        · exact (hf.comp continuous_fst).mul
            ((D_cont.comp continuous_fst).inv₀ (fun x ↦ (D_pos _).ne'))
        · exact hg.comp (continuous_snd.inv.mul continuous_fst)
      · let K := tsupport f
        have K_comp : IsCompact K := h'f
        let L := tsupport g
        have L_comp : IsCompact L := h'g
        let M := (fun (p : G × G) ↦ p.1 * p.2⁻¹) '' (K ×ˢ L)
        have M_comp : IsCompact M :=
          (K_comp.prod L_comp).image (continuous_fst.mul continuous_snd.inv)
        

Proof state:
case h'f
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulRightInvariant ν
inst✝ : IsOpenPosMeasure ν
f g : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
hg : Continuous g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
D : G → ℝ := fun x => ∫ (y : G), g (y⁻¹ * x) ∂ν
D_cont : Continuous D
D_pos : ∀ (x : G), 0 < D x
K : Set G := tsupport f
K_comp : IsCompact K
L : Set G := tsupport g
L_comp : IsCompact L
M : Set G := (fun p => p.1 * p.2⁻¹) '' K ×ˢ L
M_comp : IsCompact M
⊢ HasCompactSupport (uncurry fun x y => f x * (D x)⁻¹ * g (y⁻¹ * x))

Next tactic:
have M'_comp : IsCompact (closure M) := M_comp.closure
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts, they coincide in the following sense: they give the same value to the integral of
continuous compactly supported functions, up to a multiplicative constant. -/
@[to_additive integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport]
lemma integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ] :
    ∃ (c : ℝ≥0), ∀ (f : G → ℝ), Continuous f → HasCompactSupport f →
      ∫ x, f x ∂μ' = ∫ x, f x ∂(c • μ) := by
  -- The group has to be locally compact, otherwise all integrals vanish and the result is trivial.
  

Proof state:
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ' μ : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulLeftInvariant μ'
inst✝ : IsOpenPosMeasure μ
⊢ ∃ c, ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ

Next tactic:
by_cases H : LocallyCompactSpace G
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts, they coincide in the following sense: they give the same value to the integral of
continuous compactly supported functions, up to a multiplicative constant. -/
@[to_additive integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport]
lemma integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ] :
    ∃ (c : ℝ≥0), ∀ (f : G → ℝ), Continuous f → HasCompactSupport f →
      ∫ x, f x ∂μ' = ∫ x, f x ∂(c • μ) := by
  -- The group has to be locally compact, otherwise all integrals vanish and the result is trivial.
  by_cases H : LocallyCompactSpace G; 

Proof state:
case pos
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ' μ : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulLeftInvariant μ'
inst✝ : IsOpenPosMeasure μ
H : LocallyCompactSpace G
⊢ ∃ c, ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ
case neg
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ' μ : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulLeftInvariant μ'
inst✝ : IsOpenPosMeasure μ
H : ¬LocallyCompactSpace G
⊢ ∃ c, ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ

Next tactic:
swap
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts, they coincide in the following sense: they give the same value to the integral of
continuous compactly supported functions, up to a multiplicative constant. -/
@[to_additive integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport]
lemma integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ] :
    ∃ (c : ℝ≥0), ∀ (f : G → ℝ), Continuous f → HasCompactSupport f →
      ∫ x, f x ∂μ' = ∫ x, f x ∂(c • μ) := by
  -- The group has to be locally compact, otherwise all integrals vanish and the result is trivial.
  by_cases H : LocallyCompactSpace G; swap
  · 

Proof state:
case neg
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ' μ : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulLeftInvariant μ'
inst✝ : IsOpenPosMeasure μ
H : ¬LocallyCompactSpace G
⊢ ∃ c, ∀ (f : G → ℝ), Continuous f → HasCompactSupport f → ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂c • μ

Next tactic:
refine ⟨0, fun f f_cont f_comp ↦ ?_⟩
/-- **Uniqueness of left-invariant measures**: Given two left-invariant measures which are finite on
compacts, they coincide in the following sense: they give the same value to the integral of
continuous compactly supported functions, up to a multiplicative constant. -/
@[to_additive integral_isAddLeftInvariant_eq_smul_of_hasCompactSupport]
lemma integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport
    (μ' μ : Measure G) [IsFiniteMeasureOnCompacts μ] [IsFiniteMeasureOnCompacts μ']
    [IsMulLeftInvariant μ] [IsMulLeftInvariant μ'] [IsOpenPosMeasure μ] :
    ∃ (c : ℝ≥0), ∀ (f : G → ℝ), Continuous f → HasCompactSupport f →
      ∫ x, f x ∂μ' = ∫ x, f x ∂(c • μ) := by
  -- The group has to be locally compact, otherwise all integrals vanish and the result is trivial.
  by_cases H : LocallyCompactSpace G; swap
  · refine ⟨0, fun f f_cont f_comp ↦ ?_⟩
    

Proof state:
case neg
G : Type u_1
inst✝¹⁰ : TopologicalSpace G
inst✝⁹ : LocallyCompactSpace G
inst✝⁸ : Group G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ' μ : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts μ'
inst✝² : IsMulLeftInvariant μ
inst✝¹ : IsMulLeftInvariant μ'
inst✝ : IsOpenPosMeasure μ
H : ¬LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous f
f_comp : HasCompactSupport f
⊢ ∫ (x : G), f x ∂μ' = ∫ (x : G), f x ∂0 • μ

Next tactic:
rcases f_comp.eq_zero_or_locallyCompactSpace_of_group f_cont with hf|hf