Theorems · Theorem · measure theory

MeasureTheory.LocallyIntegrable.exists_nat_integrableOn

∀ {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} [SecondCountableTopology X],
  MeasureTheory.LocallyIntegrable f μ →
    ∃ u, (∀ (n : ℕ), IsOpen (u n)) ∧ ⋃ n, u n = Set.univ ∧ ∀ (n : ℕ), MeasureTheory.IntegrableOn f (u n) μ

If a function is locally integrable in a second countable topological space, then there exists a sequence of open sets covering the space on which it is integrable.

Defined in

Mathlib.MeasureTheory.Function.LocallyIntegrable

Cited by

1 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceTopologicalSpaceTopologicalSpaceContinuousENormSecondCountableTopology