Theorems · Theorem · measure theory

BoundedContinuousFunction.memLp_top

∀ {α : Type u_1} {E : Type u_2} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : TopologicalSpace α]
  [BorelSpace α] [inst_2 : NormedAddCommGroup E] [SecondCountableTopologyEither α E]
  (f : BoundedContinuousFunction α E), MeasureTheory.MemLp ⇑f ⊤ μ

A bounded continuous function is in L∞.

Defined in: Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions
Cited by: 1 results in Mathlib
Foundations: Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: TopologicalSpace BorelSpace NormedAddCommGroup SecondCountableTopologyEither

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Cites16

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement · cited by 62,936
TopologicalSpacestatement and proof · cited by 24,529
NormedAddCommGroupstatement and proof · cited by 15,752
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
ENNRealstatement · cited by 9,879
Top.topstatement · cited by 9,680
Filter.univ_mem'proof · cited by 1,672
BorelSpacestatement and proof · cited by 1,602
BoundedContinuousFunctionstatement and proof · cited by 511
MeasureTheory.MemLpstatement · cited by 457
ContinuousMapClass.map_continuousproof · cited by 119

Cited by1

Results whose statement or proof uses this declaration.

TemperedDistribution.MemSobolev.fourierMultiplierCLM_of_boundedproof · cited by 3