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

tactic: refine (ae_measure_preimage_mul_right_lt_top ν ν sm h3s).filter_mono ?_
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SigmaFinite ν
inst✝³ : SigmaFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
h2s : ν s ≠ 0
h3s : ν s ≠ ⊤
⊢ ∀ᵐ (x : G) ∂μ, ν ((fun y => y * x) ⁻¹' s) < ⊤
state_after: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SigmaFinite ν
inst✝³ : SigmaFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
h2s : ν s ≠ 0
h3s : ν s ≠ ⊤
⊢ ae μ ≤ ae ν

tactic: refine (absolutelyContinuous_of_isMulLeftInvariant μ ν ?_).ae_le
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SigmaFinite ν
inst✝³ : SigmaFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
h2s : ν s ≠ 0
h3s : ν s ≠ ⊤
⊢ ae μ ≤ ae 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 : SigmaFinite ν
inst✝³ : SigmaFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
h2s : ν s ≠ 0
h3s : ν s ≠ ⊤
⊢ ν ≠ 0

tactic: refine mt ?_ h2s
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SigmaFinite ν
inst✝³ : SigmaFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
h2s : ν s ≠ 0
h3s : ν s ≠ ⊤
⊢ ν ≠ 0
state_after: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SigmaFinite ν
inst✝³ : SigmaFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
h2s : ν s ≠ 0
h3s : ν s ≠ ⊤
⊢ ν = 0 → ν s = 0

tactic: intro hν
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SigmaFinite ν
inst✝³ : SigmaFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
h2s : ν s ≠ 0
h3s : ν s ≠ ⊤
⊢ ν = 0 → ν s = 0
state_after: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SigmaFinite ν
inst✝³ : SigmaFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
h2s : ν s ≠ 0
h3s : ν s ≠ ⊤
hν : ν = 0
⊢ ν s = 0

tactic: rw [hν, Measure.coe_zero, Pi.zero_apply]
state_before: G : Type u_1
inst✝⁷ : MeasurableSpace G
inst✝⁶ : Group G
inst✝⁵ : MeasurableMul₂ G
μ ν : Measure G
inst✝⁴ : SigmaFinite ν
inst✝³ : SigmaFinite μ
s : Set G
inst✝² : MeasurableInv G
inst✝¹ : μ.IsMulLeftInvariant
inst✝ : ν.IsMulLeftInvariant
sm : MeasurableSet s
h2s : ν s ≠ 0
h3s : ν s ≠ ⊤
hν : ν = 0
⊢ ν s = 0
state_after: no goals