IndepFun_iff_Indep — Mathlib

English

IndepFun f μ is equivalent to independence of the pulled-back sigmas.

Русский

IndepFun f μ эквивалентна независимости обратно-потянутых сигма-алгебр.

LaTeX

$$$\text{IndepFun}(f, μ) \iff \text{Indep}(\text{comap}(f) , μ)$$$

Lean4

theorem IndepFun_iff_Indep [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ] (f : Ω → β) (g : Ω → γ) (μ : Measure Ω) :
    IndepFun f g μ ↔ Indep (MeasurableSpace.comap f mβ) (MeasurableSpace.comap g mγ) μ := by
  simp only [IndepFun, Indep, Kernel.IndepFun]

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