Theorems · Theorem · probability

ProbabilityTheory.Kernel.setLIntegral_density

∀ {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] (a : α) {s : Set β},
      MeasurableSet s →
        ∀ {A : Set γ}, MeasurableSet A → ∫⁻ (x : γ) in A, ENNReal.ofReal (κ.density ν a x s) ∂ν a = (κ a) (A ×ˢ s)

Defined in

Mathlib.Probability.Kernel.Disintegration.Density

Cited by

2 results in Mathlib

Foundations

Depth 325 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace.CountablyGeneratedProbabilityTheory.IsFiniteKernel