Theorems · Theorem · probability

ProbabilityTheory.Kernel.tendsto_integral_density_of_monotone

∀ {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) (seq : ℕ → Set β),
      Monotone seq →
        ⋃ i, seq i = Set.univ →
          (∀ (m : ℕ), MeasurableSet (seq m)) →
            Filter.Tendsto (fun m => ∫ (x : γ), κ.density ν a x (seq m) ∂ν a) Filter.atTop (nhds ((κ a).real Set.univ))

Defined in

Mathlib.Probability.Kernel.Disintegration.Density

Cited by

2 results in Mathlib

Foundations

Depth 324 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace.CountablyGeneratedProbabilityTheory.IsFiniteKernel