Theorems · Theorem · measure theory

BoundedContinuousFunction.mem_Lp

∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {p : ENNReal} {μ : MeasureTheory.Measure α}
  [inst : TopologicalSpace α] [inst_1 : BorelSpace α] [inst_2 : NormedAddCommGroup E]
  [inst_3 : SecondCountableTopologyEither α E] [MeasureTheory.IsFiniteMeasure μ] (f : BoundedContinuousFunction α E),
  ContinuousMap.toAEEqFun μ f.toContinuousMap ∈ MeasureTheory.Lp E p μ

A bounded continuous function on a finite-measure space is in Lp.

Defined in

Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Cited by

4 results in Mathlib

Foundations

Depth 225 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceBorelSpaceNormedAddCommGroupSecondCountableTopologyEitherMeasureTheory.IsFiniteMeasure