Theorems · Theorem · measure theory

BoundedContinuousFunction.toLp_inj

∀ {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] {𝕜 : Type u_3}
  [inst_5 : Fact (1 ≤ p)] [inst_6 : NormedRing 𝕜] [inst_7 : Module 𝕜 E] [inst_8 : IsBoundedSMul 𝕜 E]
  {f g : BoundedContinuousFunction α E} [μ.IsOpenPosMeasure],
  (BoundedContinuousFunction.toLp p μ 𝕜) f = (BoundedContinuousFunction.toLp p μ 𝕜) g ↔ f = g

Defined in

Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Cited by

1 results in Mathlib

Foundations

Depth 233 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceBorelSpaceNormedAddCommGroupSecondCountableTopologyEitherMeasureTheory.IsFiniteMeasureFactNormedRingModuleIsBoundedSMulMeasureTheory.Measure.IsOpenPosMeasure