Theorems · Theorem · probability

MeasureTheory.Integrable.norm_integral_condDistrib

∀ {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}
  [inst_5 : NormedSpace ℝ F],
  AEMeasurable X μ →
    AEMeasurable Y μ →
      MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ) →
        MeasureTheory.Integrable (fun a => ‖∫ (y : Ω), f (X a, y) ∂(ProbabilityTheory.condDistrib Y X μ) (X a)‖) μ

Defined in

Mathlib.Probability.Kernel.CondDistrib

Cited by

1 results in Mathlib

Foundations

Depth 282 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceStandardBorelSpaceNonemptyNormedAddCommGroupMeasureTheory.IsFiniteMeasureNormedSpace