English
If iIndepFun f μ and each f i is AE-measurable, then the independence of the two disjoint finite index sets S and T holds.
Русский
Если f образует i-независимую систему и каждый f_i измерим/AЕ, то два дисjoint finite индекса S и T образуют независимость.
LaTeX
$$$ \\forall S,T\\; (\\,S,T\\ \\text{ Disjoint}) \\rightarrow iIndepFun f μ \\rightarrow (\\forall i, AEMeasurable (f i) μ) \\\\rightarrow \\operatorname{IndepFun} (\\lambda a (i:S) \\mapsto f i a) (\\lambda a (i:T) \\mapsto f i a) μ$$$
Lean4
/-- If `f` is a family of mutually independent random variables (`iIndepFun m f μ`) and `S, T` are
two disjoint finite index sets, then the tuple formed by `f i` for `i ∈ S` is independent of the
tuple `(f i)_i` for `i ∈ T`. -/
theorem indepFun_finset₀ (S T : Finset ι) (hST : Disjoint S T) (hf_Indep : iIndepFun f μ)
(hf_meas : ∀ i, AEMeasurable (f i) μ) : IndepFun (fun a (i : S) ↦ f i a) (fun a (i : T) ↦ f i a) μ :=
Kernel.iIndepFun.indepFun_finset₀ S T hST hf_Indep (by simp [hf_meas])
