aestronglyMeasurable_iff — Mathlib

English

If X_i are identically distributed with μ_i and u is aemeasurable, then u∘f and u∘g have the same identDistrib.

Русский

Если X_i идентично распределены и u — aemeasurable, тогда u∘f и u∘g идентично распределены.

LaTeX

$${u : γ → δ} (h : IdentDistrib f g μ ν) (hu : AEMeasurable u (Measure.map f μ)) : IdentDistrib (u ∘ f) (u ∘ g) μ ν$$

Lean4

theorem aestronglyMeasurable_iff [TopologicalSpace γ] [PseudoMetrizableSpace γ] [BorelSpace γ]
    (h : IdentDistrib f g μ ν) : AEStronglyMeasurable f μ ↔ AEStronglyMeasurable g ν :=
  ⟨fun hf => h.aestronglyMeasurable_snd hf, fun hg => h.symm.aestronglyMeasurable_snd hg⟩

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