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

tactic: refine ⟨measurable_inv, AbsolutelyContinuous.mk fun s hsm hμs => ?_⟩
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.IsMulLeftInvariant
⊢ QuasiMeasurePreserving Inv.inv 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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
⊢ (map Inv.inv μ) s = 0

tactic: rw [map_apply measurable_inv hsm, inv_preimage]
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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
⊢ (map Inv.inv μ) s = 0
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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
⊢ μ s⁻¹ = 0

tactic: have hf : 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✝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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
⊢ μ s⁻¹ = 0
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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
hf : Measurable fun z => (z.2 * z.1, z.1⁻¹)
⊢ μ s⁻¹ = 0

tactic: suffices map (fun z : G × G => (z.2 * z.1, z.1⁻¹)) (μ.prod μ) (s⁻¹ ×ˢ s⁻¹) = 0 by
  simpa only [(measurePreserving_mul_prod_inv μ μ).map_eq, prod_prod, mul_eq_zero (M₀ := ℝ≥0∞),
    or_self_iff] using 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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
hf : Measurable fun z => (z.2 * z.1, z.1⁻¹)
⊢ μ s⁻¹ = 0
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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
hf : Measurable fun z => (z.2 * z.1, z.1⁻¹)
⊢ (map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod μ)) (s⁻¹ ×ˢ s⁻¹) = 0

tactic: have hsm' : MeasurableSet (s⁻¹ ×ˢ s⁻¹) := hsm.inv.prod hsm.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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
hf : Measurable fun z => (z.2 * z.1, z.1⁻¹)
⊢ (map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod μ)) (s⁻¹ ×ˢ s⁻¹) = 0
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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
hf : Measurable fun z => (z.2 * z.1, z.1⁻¹)
hsm' : MeasurableSet (s⁻¹ ×ˢ s⁻¹)
⊢ (map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod μ)) (s⁻¹ ×ˢ s⁻¹) = 0

tactic: simp_rw [map_apply hf hsm', prod_apply_symm (μ := μ) (ν := μ) (hf hsm'), preimage_preimage,
  mk_preimage_prod, inv_preimage, inv_inv, measure_mono_null inter_subset_right hμs,
  lintegral_zero]
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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
hf : Measurable fun z => (z.2 * z.1, z.1⁻¹)
hsm' : MeasurableSet (s⁻¹ ×ˢ s⁻¹)
⊢ (map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod μ)) (s⁻¹ ×ˢ s⁻¹) = 0
state_after: no goals

tactic: simpa only [(measurePreserving_mul_prod_inv μ μ).map_eq, prod_prod, mul_eq_zero (M₀ := ℝ≥0∞),
  or_self_iff] using 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.IsMulLeftInvariant
s : Set G
hsm : MeasurableSet s
hμs : μ s = 0
hf : Measurable fun z => (z.2 * z.1, z.1⁻¹)
this : (map (fun z => (z.2 * z.1, z.1⁻¹)) (μ.prod μ)) (s⁻¹ ×ˢ s⁻¹) = 0
⊢ μ s⁻¹ = 0
state_after: no goals