Theorems · Theorem · probability

ProbabilityTheory.Kernel.setIntegral_density_of_measurableSet

∀ {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {mγ : MeasurableSpace γ} [inst : MeasurableSpace.CountablyGenerated γ] {κ : ProbabilityTheory.Kernel α (γ × β)}
  {ν : ProbabilityTheory.Kernel α γ},
  κ.fst ≤ ν →
    ∀ [ProbabilityTheory.IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β},
      MeasurableSet s → ∀ {A : Set γ}, MeasurableSet A → ∫ (x : γ) in A, κ.density ν a x s ∂ν a = (κ a).real (A ×ˢ s)

Auxiliary lemma for setIntegral_density.

Defined in

Mathlib.Probability.Kernel.Disintegration.Density

Cited by

2 results in Mathlib

Foundations

Depth 322 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace.CountablyGeneratedProbabilityTheory.IsFiniteKernel