Theorems · Theorem · measure theory

MeasureTheory.LocallyIntegrable.integrable_smul_right_of_hasCompactSupport

∀ {X : Type u_1} {E : Type u_6} [inst : MeasurableSpace X] [inst_1 : TopologicalSpace X] [inst_2 : NormedAddCommGroup E]
  {μ : MeasureTheory.Measure X} {𝕜 : Type u_9} [inst_3 : NormedRing 𝕜] [inst_4 : Module 𝕜 E] [IsBoundedSMul 𝕜 E]
  [OpensMeasurableSpace X] [T2Space X] {f : X → 𝕜},
  MeasureTheory.LocallyIntegrable f μ →
    ∀ {g : X → E}, Continuous g → HasCompactSupport g → MeasureTheory.Integrable (fun x => f x • g x) μ

If f is locally integrable and g is continuous with compact support, then f • g is integrable.

Defined in

Mathlib.MeasureTheory.Function.LocallyIntegrable

Cited by

0 results in Mathlib

Foundations

Depth 217 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceNormedAddCommGroupNormedRingModuleIsBoundedSMulOpensMeasurableSpaceT2Space