Theorems · Theorem · probability

ProbabilityTheory.Kernel.trajContent_eq_lmarginalPartialTraj

∀ {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)] {b : ℕ} {S : Set ((i : ↥(Finset.Iic b)) → X ↑i)},
  MeasurableSet S →
    ∀ (x₀ : (n : ℕ) → X n) (a : ℕ),
      (ProbabilityTheory.Kernel.trajContent κ (Preorder.frestrictLe a x₀)) (MeasureTheory.cylinder (Finset.Iic b) S) =
        ProbabilityTheory.Kernel.lmarginalPartialTraj κ a b ((MeasureTheory.cylinder (Finset.Iic b) S).indicator 1) x₀

The trajContent of a cylinder is equal to the integral of its indicator function against partialTraj.

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.Traj

Cited by

1 results in Mathlib

Foundations

Depth 237 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceProbabilityTheory.IsMarkovKernel