Theorems · Theorem · probability

ProbabilityTheory.Kernel.traj_eq_prod

∀ {X : ℕ → Type u_1} [inst : (n : ℕ) → MeasurableSpace (X n)]
  {κ : (n : ℕ) → ProbabilityTheory.Kernel ((i : ↥(Finset.Iic n)) → X ↑i) (X (n + 1))}
  [inst_1 : ∀ (n : ℕ), ProbabilityTheory.IsMarkovKernel (κ n)] (a : ℕ),
  ProbabilityTheory.Kernel.traj κ a =
    (ProbabilityTheory.Kernel.id.prod ((ProbabilityTheory.Kernel.traj κ a).map (Set.Ioi a).domRestrict)).map
      ⇑(MeasurableEquiv.IicProdIoi a)

This theorem shows that traj κ n is, up to an equivalence, the product of a deterministic kernel with another kernel. This is an intermediate result to compute integrals with respect to this kernel.

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.Traj

Cited by

1 results in Mathlib

Foundations

Depth 273 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceProbabilityTheory.IsMarkovKernel