Theorems · Theorem · probability

ProbabilityTheory.Kernel.tendsto_eLpNorm_one_densityProcess_limitProcess

∀ {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 →
        Filter.Tendsto
          (fun n =>
            MeasureTheory.eLpNorm
              ((fun x => κ.densityProcess ν n a x s) -
                MeasureTheory.Filtration.limitProcess (fun n x => κ.densityProcess ν n a x s)
                  (ProbabilityTheory.countableFiltration γ) (ν a))
              1 (ν a))
          Filter.atTop (nhds 0)

Defined in

Mathlib.Probability.Kernel.Disintegration.Density

Cited by

2 results in Mathlib

Foundations

Depth 320 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace.CountablyGeneratedProbabilityTheory.IsFiniteKernel