Theorems · Theorem · measure theory

MeasureTheory.IntegrableOn.of_ae_sdiff_eq_zero

∀ {α : Type u_1} {mα : MeasurableSpace α} {s t : Set α} {μ : MeasureTheory.Measure α} {ε' : Type u_7}
  [inst : TopologicalSpace ε'] [inst_1 : ENormedAddMonoid ε'] [TopologicalSpace.PseudoMetrizableSpace ε'] {f : α → ε'},
  MeasureTheory.IntegrableOn f s μ →
    MeasureTheory.NullMeasurableSet t μ → (∀ᵐ (x : α) ∂μ, x ∈ t \ s → f x = 0) → MeasureTheory.IntegrableOn f t μ

If a function is integrable on a set s, and its enorm vanishes on t \ s, then it is integrable on t if t is null-measurable.

Defined in

Mathlib.MeasureTheory.Integral.IntegrableOn

Cited by

4 results in Mathlib

Foundations

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

Assumes

TopologicalSpaceENormedAddMonoidTopologicalSpace.PseudoMetrizableSpace