IndepFun_iff — Mathlib

English

Independence of functions f and g is equivalent to independence of the pullback sigmas via f and g.

Русский

Независимость функций f и g эквивалентна independencia обратных сигма-алгебр через f и g.

LaTeX

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

Lean4

theorem IndepFun_iff {β γ} [mβ : MeasurableSpace β] [mγ : MeasurableSpace γ] (f : Ω → β) (g : Ω → γ) (μ : Measure Ω) :
    IndepFun f g μ ↔
      ∀ t1 t2,
        MeasurableSet[MeasurableSpace.comap f mβ] t1 →
          MeasurableSet[MeasurableSpace.comap g mγ] t2 → μ (t1 ∩ t2) = μ t1 * μ t2 :=
  by rw [IndepFun_iff_Indep, Indep_iff]

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