Theorem: MeasureTheory.lintegral_lintegral_add_neg
File: Mathlib/MeasureTheory/Group/Prod.lean

Proof steps:
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ
state_after: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹)
⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ
tactic: have h : Measurable fun z : G × G => (z.2 * z.1, z.1⁻¹) :=
    (measurable_snd.mul measurable_fst).prod_mk measurable_fst.inv
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹)
⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ
state_after: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹)
h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)
⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ
tactic: have h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν) :=
    hf.comp_quasiMeasurePreserving (measurePreserving_mul_prod_inv μ ν).quasiMeasurePreserving
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹)
h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)
⊢ ∫⁻ (x : G), ∫⁻ (y : G), f (y * x) x⁻¹ ∂ν ∂μ = ∫⁻ (x : G), ∫⁻ (y : G), f x y ∂ν ∂μ
state_after: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹)
h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)
⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν = ∫⁻ (z : G × G), f z.1 z.2 ∂μ.prod ν
tactic: simp_rw [lintegral_lintegral h2f, lintegral_lintegral hf]
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹)
h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)
⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν = ∫⁻ (z : G × G), f z.1 z.2 ∂μ.prod ν
state_after: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹)
h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)
⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν =
    ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν)
tactic: conv_rhs => rw [← (measurePreserving_mul_prod_inv μ ν).map_eq]
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹)
h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)
⊢ ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν =
    ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν)
state_after: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹)
h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)
⊢ ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν) =
    ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν
tactic: symm
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SFinite ν
inst✝³ : SFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
f : G → G → ℝ≥0∞
hf : AEMeasurable (uncurry f) (μ.prod ν)
h : Measurable fun z => (z.2 * z.1, z.1⁻¹)
h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν)
⊢ ∫⁻ (z : G × G), f z.1 z.2 ∂map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod ν) =
    ∫⁻ (z : G × G), f (z.2 * z.1) z.1⁻¹ ∂μ.prod ν
state_after: no goals
tactic: exact
    lintegral_map' (hf.mono' (measurePreserving_mul_prod_inv μ ν).map_eq.absolutelyContinuous)
      h.aemeasurable