Theorems · Theorem · probability

ProbabilityTheory.Kernel.map_frestrictLe_trajMeasure_compProd_eq_map_trajMeasure

∀ {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)] {μ₀ : MeasureTheory.Measure (X 0)}
  [MeasureTheory.IsProbabilityMeasure μ₀] {a : ℕ},
  (MeasureTheory.Measure.map (Preorder.frestrictLe a) (ProbabilityTheory.Kernel.trajMeasure μ₀ κ)).compProd (κ a) =
    MeasureTheory.Measure.map (fun x => (Preorder.frestrictLe a x, x (a + 1)))
      (ProbabilityTheory.Kernel.trajMeasure μ₀ κ)

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.Traj

Cited by

1 results in Mathlib

Foundations

Depth 277 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceProbabilityTheory.IsMarkovKernelMeasureTheory.IsProbabilityMeasure