English
For κ : Kernel α β, and measurable f : β → γ, g : β → δ, we have snd (map κ (f, g)) = map κ g provided f is measurable; if g is not measurable, the equality holds in a corrected form with measurability caveat.
Русский
Для ядра κ: Kernel α β и отображений f: β→γ, g: β→δ, равно snd (map κ (f, g)) = map κ g при измеримости f; в противном случае требуется корректировка с шансами измеримости.
LaTeX
$$$snd (map \\kappa (λ x, (f x, g x))) = map κ g$ при условии измеримости $f$$$
Lean4
theorem snd_map_prod (κ : Kernel α β) {f : β → γ} {g : β → δ} (hf : Measurable f) :
snd (map κ (fun x ↦ (f x, g x))) = map κ g :=
by
by_cases hg : Measurable g
· ext x s hs
rw [snd_apply' _ _ hs, map_apply' _ (hf.prod hg), map_apply' _ hg _ hs]
· simp only [Set.preimage, Set.mem_setOf]
· exact measurable_snd hs
· have : ¬Measurable (fun x ↦ (f x, g x)) := by contrapose! hg; exact hg.snd
simp [map_of_not_measurable _ hg, map_of_not_measurable _ this]
