Theorems · Theorem · probability

ProbabilityTheory.Kernel.lmarginalPartialTraj_succ

∀ {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 : ℕ) {f : ((n : ℕ) → X n) → ENNReal},
  Measurable f →
    ∀ (x₀ : (n : ℕ) → X n),
      ProbabilityTheory.Kernel.lmarginalPartialTraj κ a (a + 1) f x₀ =
        ∫⁻ (x : X (a + 1)), f (Function.update x₀ (a + 1) x) ∂(κ a) (Preorder.frestrictLe a x₀)

Integrating f against partialTraj κ a (a + 1) is the same as integrating against κ a.

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Cited by

1 results in Mathlib

Foundations

Depth 229 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsSFiniteKernel