ae_empty_or_univ₀ — Mathlib

English

For a quasi-ergodic map, almost-invariant measurable set s is almost surely either empty or the whole space.

Русский

Для квазирегодной карты почти инвариантное измеримое множество s почти surely либо пусто, либо всему пространству.

LaTeX

$$$\mathrm{ae\_empty\_or\_univ\ (hf, hsm, hs) : }\; s =^\mu \emptyset \lor s =^\mu \mathrm{univ}$$$

Lean4

/-- For a quasi-ergodic map, sets that are almost invariant (rather than strictly invariant) are
still either almost empty or full. -/
theorem ae_empty_or_univ₀ (hf : QuasiErgodic f μ) (hsm : NullMeasurableSet s μ) (hs : f ⁻¹' s =ᵐ[μ] s) :
    s =ᵐ[μ] (∅ : Set α) ∨ s =ᵐ[μ] univ :=
  eventuallyConst_set'.mp <| hf.aeconst_set₀ hsm hs

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