Theorems · Theorem · measure theory

BoundedContinuousFunction.Lp_norm_le

∀ {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] [inst_4 : MeasureTheory.IsFiniteMeasure μ]
  (f : BoundedContinuousFunction α E),
  ‖⟨ContinuousMap.toAEEqFun μ f.toContinuousMap, ⋯⟩‖ ≤ ↑(MeasureTheory.measureUnivNNReal μ) ^ p.toReal⁻¹ * ‖f‖

The Lp-norm of a bounded continuous function is at most a constant (depending on the measure of the whole space) times its sup-norm.

Defined in

Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Cited by

1 results in Mathlib

Foundations

Depth 227 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceBorelSpaceNormedAddCommGroupSecondCountableTopologyEitherMeasureTheory.IsFiniteMeasure