Theorems · Theorem · probability

ProbabilityTheory.Kernel.condExp_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)] {E : Type u_2} [inst_2 : NormedAddCommGroup E]
  [inst_3 : NormedSpace ℝ E] [CompleteSpace E] {a b : ℕ},
  a ≤ b →
    ∀ {x₀ : (i : ↥(Finset.Iic a)) → X ↑i} {f : ((n : ℕ) → X n) → E},
      MeasureTheory.Integrable f ((ProbabilityTheory.Kernel.traj κ a) x₀) →
        ((ProbabilityTheory.Kernel.traj κ a) x₀)[f |
            ↑MeasureTheory.Filtration.piLE b] =ᵐ[(ProbabilityTheory.Kernel.traj κ a) x₀]
          fun x => ∫ (y : (n : ℕ) → X n), f y ∂(ProbabilityTheory.Kernel.traj κ b) (Preorder.frestrictLe b x)

Defined in

Mathlib.Probability.Kernel.IonescuTulcea.Traj

Cited by

1 results in Mathlib

Foundations

Depth 299 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpaceProbabilityTheory.IsMarkovKernelNormedAddCommGroupNormedSpaceCompleteSpace