Theorems · Theorem · probability

ProbabilityTheory.setLIntegral_condKernel

∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω]
  [inst_3 : MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)}
  [inst_4 : ProbabilityTheory.IsFiniteKernel κ] {f : β × Ω → ENNReal},
  Measurable f →
    ∀ (a : α) {s : Set β},
      MeasurableSet s →
        ∀ {t : Set Ω},
          MeasurableSet t →
            ∫⁻ (b : β) in s, ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a =
              ∫⁻ (x : β × Ω) in s ×ˢ t, f x ∂κ a

Defined in

Mathlib.Probability.Kernel.Disintegration.Integral

Cited by

2 results in Mathlib

Foundations

Depth 338 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceStandardBorelSpaceNonemptyMeasurableSpace.CountableOrCountablyGeneratedProbabilityTheory.IsFiniteKernel