-- Mathlib/MeasureTheory/Measure/Haar/Unique.lean
/-- 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

Proof state:
case pos
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
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
Hf : LocallyCompactSpace G
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
M'_comp : IsCompact (closure M)
x y : G
hxy : (x, y) ∉ L ×ˢ closure M
H : x ∈ L
⊢ f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))⁻¹ * g (x, y).1 = 0

case neg
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
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
Hf : LocallyCompactSpace G
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
M'_comp : IsCompact (closure M)
x y : G
hxy : (x, y) ∉ L ×ˢ closure M
H : x ∉ L
⊢ f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))⁻¹ * g (x, y).1 = 0

Tactic:
swap

Result:
case neg
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
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
Hf : LocallyCompactSpace G
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
M'_comp : IsCompact (closure M)
x y : G
hxy : (x, y) ∉ L ×ˢ closure M
H : x ∉ L
⊢ f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))⁻¹ * g (x, y).1 = 0

case pos
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
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
Hf : LocallyCompactSpace G
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
M'_comp : IsCompact (closure M)
x y : G
hxy : (x, y) ∉ L ×ˢ closure M
H : x ∈ L
⊢ f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))⁻¹ * g (x, y).1 = 0

Proof state:
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
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
Hf : LocallyCompactSpace G
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
M'_comp : IsCompact (closure M)
x y : G
hxy : (x, y) ∉ K ×ˢ closure M
H : x ∈ K
⊢ g (y⁻¹ * x) = 0

Tactic:
apply image_eq_zero_of_nmem_tsupport

Result:
case hx
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
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
Hf : LocallyCompactSpace G
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
M'_comp : IsCompact (closure M)
x y : G
hxy : (x, y) ∉ K ×ˢ closure M
H : x ∈ K
⊢ y⁻¹ * x ∉ tsupport g

Proof state:
case hx
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
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
Hf : LocallyCompactSpace G
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
M'_comp : IsCompact (closure M)
x y : G
H : x ∈ L
hxy : y * x ∈ tsupport f
⊢ (x, y) ∈ L ×ˢ closure M

Tactic:
simp only [mem_prod, H, true_and]

Result:
case hx
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
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
Hf : LocallyCompactSpace G
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
M'_comp : IsCompact (closure M)
x y : G
H : x ∈ L
hxy : y * x ∈ tsupport f
⊢ y ∈ closure M