File: Mathlib/Probability/Kernel/Disintegration/Unique.lean

full_name: ProbabilityTheory.eq_condKernel_of_measure_eq_compProd'
code: /-- A s-finite kernel which satisfy the disintegration property of the given measure `ρ` is almost
everywhere equal to the disintegration kernel of `ρ` when evaluated on a measurable set.

This theorem in the case of finite kernels is weaker than `eq_condKernel_of_measure_eq_compProd`
which asserts that the kernels are equal almost everywhere and not just on a given measurable
set. -/
theorem eq_condKernel_of_measure_eq_compProd' (κ : Kernel α Ω) [IsSFiniteKernel κ]
    (hκ : ρ = ρ.fst ⊗ₘ κ) {s : Set Ω} (hs : MeasurableSet s) :
    ∀ᵐ x ∂ρ.fst, κ x s = ρ.condKernel x s 
goal: ∀ᵐ (x : α) ∂ρ.fst, (κ x) s = (ρ.condKernel x) s

full_name: ProbabilityTheory.eq_condKernel_of_measure_eq_compProd_real
code: /-- Auxiliary lemma for `eq_condKernel_of_measure_eq_compProd`.
Uniqueness of the disintegration kernel on ℝ. -/
lemma eq_condKernel_of_measure_eq_compProd_real {ρ : Measure (α × ℝ)} [IsFiniteMeasure ρ]
    (κ : Kernel α ℝ) [IsFiniteKernel κ] (hκ : ρ = ρ.fst ⊗ₘ κ) :
    ∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x 
goal: ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x

full_name: ProbabilityTheory.eq_condKernel_of_measure_eq_compProd
code: /-- A finite kernel which satisfies the disintegration property is almost everywhere equal to the
disintegration kernel. -/
theorem eq_condKernel_of_measure_eq_compProd (κ : Kernel α Ω) [IsFiniteKernel κ]
    (hκ : ρ = ρ.fst ⊗ₘ κ) :
    ∀ᵐ x ∂ρ.fst, κ x = ρ.condKernel x 
goal: ∀ᵐ (x : α) ∂ρ.fst, κ x = ρ.condKernel x

full_name: ProbabilityTheory.Kernel.apply_eq_measure_condKernel_of_compProd_eq
code: lemma Kernel.apply_eq_measure_condKernel_of_compProd_eq
    {ρ : Kernel α (β × Ω)} [IsFiniteKernel ρ] {κ : Kernel (α × β) Ω} [IsFiniteKernel κ]
    (hκ : Kernel.fst ρ ⊗ₖ κ = ρ) (a : α) :
    (fun b ↦ κ (a, b)) =ᵐ[Kernel.fst ρ a] (ρ a).condKernel 
goal: (fun b => κ (a, b)) =ᵐ[ρ.fst a] ⇑(ρ a).condKernel

full_name: ProbabilityTheory.Kernel.condKernel_apply_eq_condKernel
code: /-- For `fst κ a`-almost all `b`, the conditional kernel `Kernel.condKernel κ` applied to `(a, b)`
is equal to the conditional kernel of the measure `κ a` applied to `b`. -/
lemma Kernel.condKernel_apply_eq_condKernel [CountableOrCountablyGenerated α β]
    (κ : Kernel α (β × Ω)) [IsFiniteKernel κ] (a : α) :
    (fun b ↦ Kernel.condKernel κ (a, b)) =ᵐ[Kernel.fst κ a] (κ a).condKernel 
goal: (fun b => κ.condKernel (a, b)) =ᵐ[κ.fst a] ⇑(κ a).condKernel

full_name: ProbabilityTheory.condKernel_const
code: lemma condKernel_const [CountableOrCountablyGenerated α β] (ρ : Measure (β × Ω)) [IsFiniteMeasure ρ]
    (a : α) :
    (fun b ↦ Kernel.condKernel (Kernel.const α ρ) (a, b)) =ᵐ[ρ.fst] ρ.condKernel 
goal: (fun b => (ProbabilityTheory.Kernel.const α ρ).condKernel (a, b)) =ᵐ[ρ.fst] ⇑ρ.condKernel

full_name: ProbabilityTheory.eq_condKernel_of_kernel_eq_compProd
code: /-- A finite kernel which satisfies the disintegration property is almost everywhere equal to the
disintegration kernel. -/
theorem eq_condKernel_of_kernel_eq_compProd [CountableOrCountablyGenerated α β]
    {ρ : Kernel α (β × Ω)} [IsFiniteKernel ρ] {κ : Kernel (α × β) Ω} [IsFiniteKernel κ]
    (hκ : Kernel.fst ρ ⊗ₖ κ = ρ) (a : α) :
    ∀ᵐ x ∂(Kernel.fst ρ a), κ (a, x) = Kernel.condKernel ρ (a, x) 
goal: ∀ᵐ (x : β) ∂ρ.fst a, κ (a, x) = ρ.condKernel (a, x)