Theorems · Theorem · measure theory

MeasureTheory.Measure.FiniteAtFilter.integrableAtFilter

∀ {α : Type u_1} {E : Type u_5} {mα : MeasurableSpace α} [inst : NormedAddCommGroup E] {μ : MeasureTheory.Measure α}
  {f : α → E} {l : Filter α} [l.IsMeasurablyGenerated],
  StronglyMeasurableAtFilter f l μ →
    μ.FiniteAtFilter l →
      Filter.IsBoundedUnder (fun x1 x2 => x1 ≤ x2) l (norm ∘ f) → MeasureTheory.IntegrableAtFilter f l μ

If μ is a measure finite at filter l and f is a function such that its norm is bounded above at l, then f is integrable at l.

Defined in

Mathlib.MeasureTheory.Integral.IntegrableOn

Cited by

2 results in Mathlib

Foundations

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

Assumes

NormedAddCommGroupFilter.IsMeasurablyGenerated