measurable_of_restrict_of_restrict_compl

English

If f is measurable on s and on sᶜ, then f is measurable on the whole space.

Русский

Если f измеримо на s и на дополнении sᶜ, тогда f измерим на всей пространстве.

LaTeX

$$$\\\\operatorname{Measurable} f \\\\Leftarrow \\\\operatorname{Measurable}(s\\\\,f) \\\\land \\\\operatorname{Measurable}(s^c\\\\,f)$$$

Lean4

theorem measurable_of_restrict_of_restrict_compl {f : α → β} {s : Set α} (hs : MeasurableSet s)
    (h₁ : Measurable (s.restrict f)) (h₂ : Measurable (sᶜ.restrict f)) : Measurable f :=
  measurable_of_measurable_union_cover s sᶜ hs hs.compl (union_compl_self s).ge h₁ h₂

Похожие утверждения