Theorems · Theorem · measure theory

ContinuousOn.integrableOn_of_subset_isCompact

∀ {X : Type u_1} {E : Type u_6} [inst : MeasurableSpace X] [inst_1 : TopologicalSpace X] [inst_2 : NormedAddCommGroup E]
  {μ : MeasureTheory.Measure X} {s : Set X} [OpensMeasurableSpace X] {K : Set X} {f : X → E},
  ContinuousOn f K → IsCompact K → MeasurableSet s → s ⊆ K → μ s ≠ ⊤ → MeasureTheory.IntegrableOn f s μ

If f is continuous on a compact set K, then it is integrable on any measurable subset s ⊆ K of finite measure.

Defined in

Mathlib.MeasureTheory.Function.LocallyIntegrable

Cited by

3 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceNormedAddCommGroupOpensMeasurableSpace