Theorems · Theorem · probability

ProbabilityTheory.Kernel.lmarginalPartialTraj_eq_lintegral_map

∀ {X : ℕ → Type u_1} {mX : (n : ℕ) → MeasurableSpace (X n)} {a b : ℕ}
  {κ : (n : ℕ) → ProbabilityTheory.Kernel ((i : ↥(Finset.Iic n)) → X ↑i) (X (n + 1))}
  [∀ (n : ℕ), ProbabilityTheory.IsSFiniteKernel (κ n)] {f : ((n : ℕ) → X n) → ENNReal},
  Measurable f →
    ∀ (x₀ : (n : ℕ) → X n),
      ProbabilityTheory.Kernel.lmarginalPartialTraj κ a b f x₀ =
        ∫⁻ (x : (i : ↥(Finset.Ioc a b)) → X ↑i),
          f
            (Function.updateFinset x₀ (Finset.Ioc a b)
              x) ∂((ProbabilityTheory.Kernel.partialTraj κ a b).map (Finset.restrict₂ ⋯)) (Preorder.frestrictLe a x₀)

Integrating f against partialTraj κ a b x is the same as integrating only over the variables from x_{a+1} to x_b.

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj

Cited by

2 results in Mathlib

Foundations

Depth 232 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ProbabilityTheory.IsSFiniteKernel