Theorems · Theorem · probability

ProbabilityTheory.Kernel.condExp_densityProcess

∀ {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] {i j : ℕ},
      i ≤ j →
        ∀ (a : α) {s : Set β},
          MeasurableSet s →
            (ν a)[fun x => κ.densityProcess ν j a x s | ↑(ProbabilityTheory.countableFiltration γ) i] =ᵐ[ν a] fun x =>
              κ.densityProcess ν i a x s

Defined in

Mathlib.Probability.Kernel.Disintegration.Density

Cited by

1 results in Mathlib

Foundations

Depth 299 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace.CountablyGeneratedProbabilityTheory.IsFiniteKernel