∀ {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} (x₀ : (i : ↥(Finset.Iic a)) → X ↑i)
  {f : ((n : ℕ) → X n) → ENNReal},
  AEMeasurable f ((ProbabilityTheory.Kernel.traj κ a) x₀) →
    ∫⁻ (x : (n : ℕ) → X n), f x ∂(ProbabilityTheory.Kernel.traj κ a) x₀ =
      ∫⁻ (x : (n : ℕ) → X n), f (Function.updateFinset x (Finset.Iic a) x₀) ∂(ProbabilityTheory.Kernel.traj κ a) x₀