lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun''

English

The independence of the family X_i is equivalent to the equality of the joint distribution under μ with the infinite product of marginals, assuming measurability of each X_i.

Русский

Независимость семейства X_i эквивалентна равенству совместного распределения и произведения мажорных мер, при условии измеримости каждого X_i.

LaTeX

$$$iIndepFun X μ \iff μ.map (\lambda ω i. X_i ω) = infinitePi (\lambda i. μ.map (X_i))$$$

Lean4

theorem lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' (h_meas_f : AEMeasurable f μ)
    (h_meas_g : AEMeasurable g μ) (h_indep_fun : IndepFun f g μ) : ∫⁻ ω, f ω * g ω ∂μ = (∫⁻ ω, f ω ∂μ) * ∫⁻ ω, g ω ∂μ :=
  lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun' h_meas_f h_meas_g h_indep_fun

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