Theorems · Theorem · measure theory

ContinuousMap.hasSum_of_hasSum_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] [inst_4 : CompactSpace α] [inst_5 : MeasureTheory.IsFiniteMeasure μ]
  {𝕜 : Type u_3} [inst_6 : Fact (1 ≤ p)] [inst_7 : NormedRing 𝕜] [inst_8 : Module 𝕜 E] [inst_9 : IsBoundedSMul 𝕜 E]
  {β : Type u_4} [μ.IsOpenPosMeasure] {g : β → C(α, E)} {f : C(α, E)},
  Summable g → HasSum (⇑(ContinuousMap.toLp p μ 𝕜) ∘ g) ((ContinuousMap.toLp p μ 𝕜) f) → HasSum g f

If a sum of continuous functions g n is convergent, and the same sum converges in Lᵖ to h, then in fact g n converges uniformly to h.

Defined in

Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions

Cited by

2 results in Mathlib

Foundations

Depth 236 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceBorelSpaceNormedAddCommGroupSecondCountableTopologyEitherCompactSpaceMeasureTheory.IsFiniteMeasureFactNormedRingModuleIsBoundedSMulMeasureTheory.Measure.IsOpenPosMeasure