Theorems · Theorem · measure theory

MeasureTheory.ae_eq_zero_of_forall_setIntegral_eq_of_finStronglyMeasurable_trim

∀ {α : Type u_1} {E : Type u_2} {m m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup E]
  [inst_1 : NormedSpace ℝ E] [CompleteSpace E] (hm : m ≤ m0) {f : α → E},
  (∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ) →
    (∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0) →
      MeasureTheory.FinStronglyMeasurable f (μ.trim hm) → f =ᵐ[μ] 0

Defined in

Mathlib.MeasureTheory.Function.AEEqOfIntegral

Cited by

1 results in Mathlib

Foundations

Depth 262 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpace