Theorems · Theorem · measure theory

MeasureTheory.ae_eq_condExp_of_forall_setIntegral_eq

∀ {α : Type u_1} {E : Type u_3} {m m₀ : MeasurableSpace α} {μ : MeasureTheory.Measure α} [inst : NormedAddCommGroup E]
  [inst_1 : NormedSpace ℝ E] [CompleteSpace E] (hm : m ≤ m₀) [MeasureTheory.SigmaFinite (μ.trim hm)] {f g : α → E},
  MeasureTheory.Integrable f μ →
    (∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn g s μ) →
      (∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, f x ∂μ) →
        MeasureTheory.AEStronglyMeasurable g μ → g =ᵐ[μ] μ[f | m]

Uniqueness of the conditional expectation If a function is a.e. m-measurable, verifies an integrability condition and has same integral as f on all m-measurable sets, then it is a.e. equal to μ[f|hm].

Defined in

Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic

Cited by

12 results in Mathlib

Foundations

Depth 298 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceCompleteSpaceMeasureTheory.SigmaFinite