English
Under finite measure, independence of f and g implies equality of the pushforward maps of their product with the monoidal convolution mconv.
Русский
При сколькими конечными мерами независимость f и g порождает равенство образующих мер от произведения через конволюцию.
LaTeX
$$$\\map(f\\cdot g) = \\map f \\;\\mconv\\; \\map g$ (for independent $f,g$ with finite measure).$$
Lean4
/-- Two random variables are independent if and only if their joint characteristic function is equal
to the product of the characteristic functions. This is the version for Hilbert spaces, see
`indepFun_iff_charFunDual_prod` for the Banach space version. -/
theorem indepFun_iff_charFun_prod (hX : AEMeasurable X P) (hY : AEMeasurable Y P) :
IndepFun X Y P ↔
∀ t, charFun (P.map (fun ω ↦ toLp 2 (X ω, Y ω))) t = charFun (P.map X) t.ofLp.1 * charFun (P.map Y) t.ofLp.2 :=
by
rw [indepFun_iff_map_prod_eq_prod_map_map hX hY, ← charFun_eq_prod_iff,
AEMeasurable.map_map_of_aemeasurable (by fun_prop) (by fun_prop), Function.comp_def]
