English
For countable index set, the iIndepFun X μ is equivalent to the product measure equality μ.map (X i) for all i against the infinitePi measure, under AEMeasurability assumptions.
Русский
Для счётного индексового множества iIndepFun эквивалентно равенству мер в произведении terhadap бесконечномуPi, в условиях AEMеасумируемости.
LaTeX
$$$\text{Countable ι} \Rightarrow\; iIndepFun X μ \iff μ.map (λ ω i. X_i ω) = infinitePi (λ i. μ.map (X_i)).$$$
Lean4
/-- Random variables are independent iff their joint distribution is the product measure. This is
an `AEMeasurable` version of `iIndepFun_iff_map_fun_eq_infinitePi_map`, which is why it requires
`ι` to be countable. -/
theorem iIndepFun_iff_map_fun_eq_infinitePi_map₀' [Countable ι] (mX : ∀ i, AEMeasurable (X i) μ) :
haveI _ i := isProbabilityMeasure_map (mX i)
iIndepFun X μ ↔ μ.map (fun ω i ↦ X i ω) = infinitePi (fun i ↦ μ.map (X i)) :=
iIndepFun_iff_map_fun_eq_infinitePi_map₀ <| aemeasurable_pi_iff.2 mX
