Theorems · Theorem · measure theory

MeasureTheory.lintegral_eq_zero_iff

∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal},
  Measurable f → (∫⁻ (a : α), f a ∂μ = 0 ↔ f =ᵐ[μ] 0)

The measurability assumption is necessary, otherwise there are counterexamples: for instance, the conclusion fails if f is the characteristic function of a Vitali set.

Defined in

Mathlib.MeasureTheory.Integral.Lebesgue.Basic

Cited by

22 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound