Theorems · Theorem · measure theory

ContinuousOn.stronglyMeasurableAtFilter_nhdsWithin

∀ {α : Type u_7} {β : Type u_8} [inst : MeasurableSpace α] [inst_1 : TopologicalSpace α] [OpensMeasurableSpace α]
  [inst_3 : TopologicalSpace β] [TopologicalSpace.PseudoMetrizableSpace β] [SecondCountableTopologyEither α β]
  {f : α → β} {s : Set α} {μ : MeasureTheory.Measure α},
  ContinuousOn f s → MeasurableSet s → ∀ (x : α), StronglyMeasurableAtFilter f (nhdsWithin x s) μ

If a function is continuous on a measurable set s, then it is measurable at the filter 𝓝[s] x for all x.

Defined in

Mathlib.MeasureTheory.Integral.IntegrableOn

Cited by

2 results in Mathlib

Foundations

Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceOpensMeasurableSpaceTopologicalSpaceTopologicalSpace.PseudoMetrizableSpaceSecondCountableTopologyEither