Theorems · Theorem · measure theory

ContinuousOn.locallyIntegrableOn

∀ {X : Type u_1} {E : Type u_6} [inst : MeasurableSpace X] [inst_1 : TopologicalSpace X] [inst_2 : NormedAddCommGroup E]
  {μ : MeasureTheory.Measure X} [OpensMeasurableSpace X] {K : Set X} {f : X → E}
  [MeasureTheory.IsLocallyFiniteMeasure μ] [SecondCountableTopologyEither X E],
  ContinuousOn f K → MeasurableSet K → MeasureTheory.LocallyIntegrableOn f K μ

A function f continuous on a set K is locally integrable on this set with respect to any locally finite measure.

Defined in

Mathlib.MeasureTheory.Function.LocallyIntegrable

Cited by

3 results in Mathlib

Foundations

Depth 208 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceNormedAddCommGroupOpensMeasurableSpaceMeasureTheory.IsLocallyFiniteMeasureSecondCountableTopologyEither