Theorems · Theorem · measure theory

MeasureTheory.Lp.mem_boundedContinuousFunction_iff

∀ {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] {f : ↥(MeasureTheory.Lp E p μ)},
  f ∈ MeasureTheory.Lp.boundedContinuousFunction E p μ ↔ ∃ f₀, ContinuousMap.toAEEqFun μ f₀.toContinuousMap = ↑f

By definition, the elements of Lp.boundedContinuousFunction E p μ are the elements of Lp E p μ which contain a bounded continuous representative.

Defined in

Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Cited by

0 results in Mathlib

Foundations

Depth 222 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceBorelSpaceNormedAddCommGroupSecondCountableTopologyEither