Theorems · Theorem · probability

ProbabilityTheory.Kernel.trajContent_cylinder

∀ {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 b : ℕ} {S : Set ((i : ↥(Finset.Iic b)) → X ↑i)},
  MeasurableSet S →
    ∀ (x₀ : (i : ↥(Finset.Iic a)) → X ↑i),
      (ProbabilityTheory.Kernel.trajContent κ x₀) (MeasureTheory.cylinder (Finset.Iic b) S) =
        ((ProbabilityTheory.Kernel.partialTraj κ a b) x₀) S

The trajContent κ x₀ of a cylinder indexed by first coordinates is given by partialTraj.

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.Traj

Cited by

2 results in Mathlib

Foundations

Depth 236 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceProbabilityTheory.IsMarkovKernel