Theorems · Theorem · measure theory

MeasureTheory.IntegrableOn.integrable_of_forall_notMem_eq_zero

∀ {α : Type u_1} {mα : MeasurableSpace α} {s : 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 → (∀ x ∉ s, f x = 0) → MeasureTheory.Integrable f μ

If a function is integrable on a set s and vanishes everywhere on its complement, then it is integrable.

Defined in

Mathlib.MeasureTheory.Integral.IntegrableOn

Cited by

2 results in Mathlib

Foundations

Depth 214 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceENormedAddMonoidTopologicalSpace.PseudoMetrizableSpace