case intro.intro.intro
α : Type u_1
m : MeasurableSpace α
f : α → α
s : Set α
μ : MeasureTheory.Measure α
hf : QuasiErgodic f
hs : MeasurableSet s
hs' : f ⁻¹' s =ᶠ[ae μ] s
t : Set α
h₀ : MeasurableSet t
h₁ : t =ᶠ[ae μ] s
h₂ : f ⁻¹' t = t
⊢ s =ᶠ[ae μ] ∅ ∨ s =ᶠ[ae μ] univ
α : Type u_1
m : MeasurableSpace α
f : α → α
s : Set α
μ : MeasureTheory.Measure α
hf : QuasiErgodic f
hs : MeasurableSet s
hs' : f ⁻¹' s =ᶠ[ae μ] s
⊢ s =ᶠ[ae μ] ∅ ∨ s =ᶠ[ae μ] univ
obtain ⟨t, h₀, h₁, h₂⟩ := hf.toQuasiMeasurePreserving.exists_preimage_eq_of_preimage_ae hs hs'
no goals
case intro.intro.intro
α : Type u_1
m : MeasurableSpace α
f : α → α
s : Set α
μ : MeasureTheory.Measure α
hf : QuasiErgodic f
hs : MeasurableSet s
hs' : f ⁻¹' s =ᶠ[ae μ] s
t : Set α
h₀ : MeasurableSet t
h₁ : t =ᶠ[ae μ] s
h₂ : f ⁻¹' t = t
⊢ s =ᶠ[ae μ] ∅ ∨ s =ᶠ[ae μ] univ
rcases hf.ae_empty_or_univ h₀ h₂ with (h₃ | h₃) <;> [left; right] <;> exact ae_eq_trans h₁.symm h₃