Theorems · Theorem · measure theory

ContinuousOn.aestronglyMeasurable

∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β]
  [h : SecondCountableTopologyEither α β] [OpensMeasurableSpace α] [TopologicalSpace.PseudoMetrizableSpace β]
  {f : α → β} {s : Set α} {μ : MeasureTheory.Measure α},
  ContinuousOn f s → MeasurableSet s → MeasureTheory.AEStronglyMeasurable f (μ.restrict s)

A function which is continuous on a set s is almost everywhere strongly measurable with respect to μ.restrict s when either the source space or the target space is second-countable.

Defined in

Mathlib.MeasureTheory.Integral.IntegrableOn

Cited by

21 results in Mathlib

Foundations

Depth 201 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceTopologicalSpaceSecondCountableTopologyEitherOpensMeasurableSpaceTopologicalSpace.PseudoMetrizableSpace