Theorems · Theorem · probability

ProbabilityTheory.Kernel.partialTraj_eq_prod

∀ {X : ℕ → Type u_1} {mX : (n : ℕ) → MeasurableSpace (X n)}
  {κ : (n : ℕ) → ProbabilityTheory.Kernel ((i : ↥(Finset.Iic n)) → X ↑i) (X (n + 1))}
  [∀ (n : ℕ), ProbabilityTheory.IsSFiniteKernel (κ n)] (a b : ℕ),
  ProbabilityTheory.Kernel.partialTraj κ a b =
    (ProbabilityTheory.Kernel.id.prod ((ProbabilityTheory.Kernel.partialTraj κ a b).map (Finset.restrict₂ ⋯))).map
      (IicProdIoc a b)

This is a technical lemma saying that partialTraj κ a b consists of two independent parts, the first one being the identity. It allows to compute integrals.

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Cited by

3 results in Mathlib

Foundations

Depth 231 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsSFiniteKernel