eq_const_of_compMeasurePreserving_eq

English

Let f be ergodic and g be a measure-valued function; if g after composition with f equals g, then g is constant a.e.

Русский

Пусть f эргодичен, а g — мера-значение функция; если g ◦ f = g почти наверняка, тогда g константна почти всюду.

LaTeX

$$$[Ergodic(f, \\\\mu)] \\Rightarrow \\exists c, g = \\\\text{const}(α, c)$$$

Lean4

theorem eq_const_of_compMeasurePreserving_eq (h : Ergodic f μ) {g : α →ₘ[μ] X}
    (hg_eq : g.compMeasurePreserving f h.1 = g) : ∃ c, g = .const α c :=
  h.quasiErgodic.eq_const_of_compQuasiMeasurePreserving_eq hg_eq

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