Theorems · Theorem · measure theory

MeasureTheory.IntegrableOn.continuousOn_smul_of_subset

∀ {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] {A K : Set X} {𝕜 : Type u_9} [inst_4 : NormedRing 𝕜]
  [inst_5 : Module 𝕜 E] [IsBoundedSMul 𝕜 E] [SecondCountableTopologyEither X 𝕜] {f : X → 𝕜},
  ContinuousOn f K →
    ∀ {g : X → E},
      MeasureTheory.IntegrableOn g A μ →
        IsCompact K → MeasurableSet A → A ⊆ K → MeasureTheory.IntegrableOn (fun x => f x • g x) A μ

Defined in

Mathlib.MeasureTheory.Function.LocallyIntegrable

Cited by

2 results in Mathlib

Foundations

Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceNormedAddCommGroupOpensMeasurableSpaceNormedRingModuleIsBoundedSMulSecondCountableTopologyEither