Theorems · Theorem · probability

ProbabilityTheory.Kernel.setIntegral_densityProcess_of_mem

∀ {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 →
    ∀ [hν : ProbabilityTheory.IsFiniteKernel ν] (n : ℕ) (a : α) {s : Set β},
      MeasurableSet s →
        ∀ {u : Set γ},
          u ∈ MeasurableSpace.countablePartition γ n →
            ∫ (x : γ) in u, κ.densityProcess ν n a x s ∂ν a = (κ a).real (u ×ˢ s)

Defined in

Mathlib.Probability.Kernel.Disintegration.Density

Cited by

1 results in Mathlib

Foundations

Depth 255 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace.CountablyGeneratedProbabilityTheory.IsFiniteKernel