∀ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  {κ : ProbabilityTheory.Kernel α (β × ℝ)} {ν : ProbabilityTheory.Kernel α β} {f : α × β → StieltjesFunction ℝ},
  ProbabilityTheory.IsCondKernelCDF f κ ν →
    ∀ (a : α) (x : ℝ), ∫ (b : β), ↑(f (a, b)) x ∂ν a = (κ a).real (Set.univ ×ˢ Set.Iic x)