∀ {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
  [inst : MeasurableSpace Ω] [inst_1 : StandardBorelSpace Ω] [inst_2 : Nonempty Ω]
  {ρ : ProbabilityTheory.Kernel α (β × Ω)} [inst_3 : ProbabilityTheory.IsFiniteKernel ρ]
  {κ : ProbabilityTheory.Kernel (α × β) Ω} [ProbabilityTheory.IsFiniteKernel κ],
  ρ.fst.compProd κ = ρ → ∀ (a : α), (fun b => κ (a, b)) =ᵐ[ρ.fst a] ⇑(ρ a).condKernel