English
Let f: X → Z be a surjective Borel-measurable map from a standard Borel space X onto a countably separated measurable space Z. For any measurable space β and any g: Z → β, the composition g ∘ f is measurable if and only if g is measurable.
Русский
Пусть f: X → Z — сюръективная бореллерова отображение из стандартного пространства Бореля X на счётно отделимое učinkov пространство Z. Для любого измеримого пространства β и любого g: Z → β, композиция g ∘ f измерима тогда и только тогда, когда измерима g.
LaTeX
$$$\\operatorname{Measurable}(g \\circ f) \\iff \\operatorname{Measurable}(g)$$$
Lean4
/-- If `f : X → Z` is a surjective Borel measurable map from a standard Borel space
to a countably separated measurable space,
then for any measurable space `α` and `g : Z → α`, the composition
`g ∘ f` is measurable if and only if `g` is measurable. -/
theorem measurable_comp_iff_of_surjective [CountablySeparated Z] {f : X → Z} (hf : Measurable f) (hsurj : Surjective f)
{g : Z → β} : Measurable (g ∘ f) ↔ Measurable g :=
forall₂_congr fun s _ => measurableSet_preimage_iff_of_surjective hf hsurj (s := g ⁻¹' s)
