Theorems · Theorem · probability

ProbabilityTheory.Kernel.tendsto_densityProcess_atTop_empty_of_antitone

∀ {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) [ProbabilityTheory.IsFiniteKernel κ] (n : ℕ) (a : α) (x : γ) (seq : ℕ → Set β),
  Antitone seq →
    ⋂ i, seq i = ∅ →
      (∀ (m : ℕ), MeasurableSet (seq m)) →
        Filter.Tendsto (fun m => κ.densityProcess ν n a x (seq m)) Filter.atTop (nhds (κ.densityProcess ν n a x ∅))

Defined in

Mathlib.Probability.Kernel.Disintegration.Density

Cited by

1 results in Mathlib

Foundations

Depth 203 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace.CountablyGeneratedProbabilityTheory.IsFiniteKernel