Theorems · Theorem · measure theory

MeasureTheory.locallyIntegrableOn_iff

∀ {X : Type u_1} {ε : Type u_3} [inst : MeasurableSpace X] [inst_1 : TopologicalSpace X] [inst_2 : TopologicalSpace ε]
  [inst_3 : ContinuousENorm ε] {f : X → ε} {μ : MeasureTheory.Measure X} {s : Set X}
  [TopologicalSpace.PseudoMetrizableSpace ε] [LocallyCompactSpace X],
  IsLocallyClosed s →
    (MeasureTheory.LocallyIntegrableOn f s μ ↔ ∀ k ⊆ s, IsCompact k → MeasureTheory.IntegrableOn f k μ)

If s is locally closed (e.g. open or closed), then f is locally integrable on s iff it is integrable on every compact subset contained in s.

Defined in

Mathlib.MeasureTheory.Function.LocallyIntegrable

Cited by

5 results in Mathlib

Foundations

Depth 212 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceTopologicalSpaceContinuousENormTopologicalSpace.PseudoMetrizableSpaceLocallyCompactSpace