Theorems · Theorem · probability

ProbabilityTheory.Kernel.eq_traj

∀ {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 : ℕ}
  (η : ProbabilityTheory.Kernel ((i : ↥(Finset.Iic a)) → X ↑i) ((n : ℕ) → X n)),
  (∀ (b : ℕ), η.map (Preorder.frestrictLe b) = ProbabilityTheory.Kernel.partialTraj κ a b) →
    η = ProbabilityTheory.Kernel.traj κ a

To check that η = traj κ a it is enough to show that the restriction of η to variables ≤ b is partialTraj κ a b.

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.Traj

Cited by

1 results in Mathlib

Foundations

Depth 272 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceProbabilityTheory.IsMarkovKernel