stronglyMeasurable_of_restrict_of_restrict_compl

English

An extended induction principle for strongly measurable functions using union covers and piecewise extensions.

Русский

Усиленная индукционная слабость для сильно измеримых функций через объединение по покрытиям и piecewise расширения.

LaTeX

$$$\text{induction'}(\text{P})$ expresses that constants, piecewise constructions, and limits suffice to prove P for all strongly measurable functions.$$

Lean4

theorem _root_.stronglyMeasurable_of_restrict_of_restrict_compl {_ : MeasurableSpace α} [TopologicalSpace β] {f : α → β}
    {s : Set α} (hs : MeasurableSet s) (h₁ : StronglyMeasurable (s.restrict f))
    (h₂ : StronglyMeasurable (sᶜ.restrict f)) : StronglyMeasurable f :=
  stronglyMeasurable_of_stronglyMeasurable_union_cover s sᶜ hs hs.compl (union_compl_self s).ge h₁ h₂

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