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

tactic: have :=
  (quasiMeasurePreserving_mul_right μ.inv g⁻¹).mono (inv_absolutelyContinuous μ.inv)
    (absolutelyContinuous_inv μ.inv)
state_before: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SigmaFinite ν
inst✝² : SigmaFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
⊢ QuasiMeasurePreserving (fun h => g * h) μ μ
state_after: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SigmaFinite ν
inst✝² : SigmaFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this : QuasiMeasurePreserving (fun h => h * g⁻¹) μ.inv.inv μ.inv.inv
⊢ QuasiMeasurePreserving (fun h => g * h) μ μ

tactic: rw [μ.inv_inv] at this
state_before: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SigmaFinite ν
inst✝² : SigmaFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this : QuasiMeasurePreserving (fun h => h * g⁻¹) μ.inv.inv μ.inv.inv
⊢ QuasiMeasurePreserving (fun h => g * h) μ μ
state_after: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SigmaFinite ν
inst✝² : SigmaFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this : QuasiMeasurePreserving (fun h => h * g⁻¹) μ μ
⊢ QuasiMeasurePreserving (fun h => g * h) μ μ

tactic: have :=
  (quasiMeasurePreserving_inv_of_right_invariant μ).comp
    (this.comp (quasiMeasurePreserving_inv_of_right_invariant μ))
state_before: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SigmaFinite ν
inst✝² : SigmaFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this : QuasiMeasurePreserving (fun h => h * g⁻¹) μ μ
⊢ QuasiMeasurePreserving (fun h => g * h) μ μ
state_after: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SigmaFinite ν
inst✝² : SigmaFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this✝ : QuasiMeasurePreserving (fun h => h * g⁻¹) μ μ
this : QuasiMeasurePreserving (Inv.inv ∘ (fun h => h * g⁻¹) ∘ Inv.inv) μ μ
⊢ QuasiMeasurePreserving (fun h => g * h) μ μ

tactic: simp_rw [Function.comp, mul_inv_rev, inv_inv] at this
state_before: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SigmaFinite ν
inst✝² : SigmaFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this✝ : QuasiMeasurePreserving (fun h => h * g⁻¹) μ μ
this : QuasiMeasurePreserving (Inv.inv ∘ (fun h => h * g⁻¹) ∘ Inv.inv) μ μ
⊢ QuasiMeasurePreserving (fun h => g * h) μ μ
state_after: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SigmaFinite ν
inst✝² : SigmaFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this✝ : QuasiMeasurePreserving (fun h => h * g⁻¹) μ μ
this : QuasiMeasurePreserving (fun x => g * x) μ μ
⊢ QuasiMeasurePreserving (fun h => g * h) μ μ

tactic: exact this
state_before: G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SigmaFinite ν
inst✝² : SigmaFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
g : G
this✝ : QuasiMeasurePreserving (fun h => h * g⁻¹) μ μ
this : QuasiMeasurePreserving (fun x => g * x) μ μ
⊢ QuasiMeasurePreserving (fun h => g * h) μ μ
state_after: no goals