Theorems · Theorem · probability

MeasureTheory.AEStronglyMeasurable.ae_integrable_condDistrib_map_iff

∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {F : Type u_4} [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω]
  [inst_2 : Nonempty Ω] [inst_3 : NormedAddCommGroup F] {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α}
  [inst_4 : MeasureTheory.IsFiniteMeasure μ] {X : α → β} {Y : α → Ω} {mβ : MeasurableSpace β} {f : β × Ω → F},
  AEMeasurable Y μ →
    MeasureTheory.AEStronglyMeasurable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) →
      ((∀ᵐ (a : β) ∂MeasureTheory.Measure.map X μ,
            MeasureTheory.Integrable (fun ω => f (a, ω)) ((ProbabilityTheory.condDistrib Y X μ) a)) ∧
          MeasureTheory.Integrable (fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂(ProbabilityTheory.condDistrib Y X μ) a)
            (MeasureTheory.Measure.map X μ) ↔
        MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ))

Defined in

Mathlib.Probability.Kernel.CondDistrib

Cited by

0 results in Mathlib

Foundations

Depth 280 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceStandardBorelSpaceNonemptyNormedAddCommGroupMeasureTheory.IsFiniteMeasure