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namespace OAI
section
section
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped ENNReal
namespace HardSphere.Comparison
private theorem exists_kernel_pi_fin {α : Type*} [MeasurableSpace α] (n : ℕ)
(β : Fin n → Type*) [∀ i, MeasurableSpace (β i)]
(κ : ∀ i, Kernel α (β i)) [∀ i, IsFiniteKernel (κ i)] :
∃ η : Kernel α (∀ i, β i), IsFiniteKernel η ∧
∀ a, η a = Measure.pi (fun i ↦ κ i a) := by
induction n with
| zero =>
let η : Kernel α (∀ i, β i) := Kernel.deterministic (fun _ i ↦ Fin.elim0 i) measurable_const
refine ⟨η, inferInstance, fun a ↦ ?_⟩
rw [Kernel.deterministic_apply, Measure.pi_of_empty (fun i ↦ κ i a) (fun i ↦ Fin.elim0 i)]
| succ n ih =>
obtain ⟨η, hη, heq⟩ := ih (fun i ↦ β ((0 : Fin (n+1)).succAbove i))
(fun i ↦ κ ((0 : Fin (n+1)).succAbove i))
let := hη
let e := (MeasurableEquiv.piFinSuccAbove β 0).symm
refine ⟨((κ 0).prod η).map e, inferInstance, fun a ↦ ?_⟩
rw [Kernel.map_apply _ e.measurable, Kernel.prod_apply, heq]
exact ((measurePreserving_piFinSuccAbove (fun i ↦ κ i a) 0).symm
(MeasurableEquiv.piFinSuccAbove β 0)).map_eq
theorem exists_kernel_pi {α ι : Type*} [MeasurableSpace α] [Fintype ι]
(β : ι → Type*) [∀ i, MeasurableSpace (β i)]
(κ : ∀ i, Kernel α (β i)) [∀ i, IsFiniteKernel (κ i)] :
∃ η : Kernel α (∀ i, β i), IsFiniteKernel η ∧
∀ a, η a = Measure.pi (fun i ↦ κ i a) := by
classical
let e := (Fintype.equivFin ι).symm
obtain ⟨η, hη, heq⟩ := exists_kernel_pi_fin (Fintype.card ι)
(fun i ↦ β (e i)) (fun i ↦ κ (e i))
let := hη
let E := MeasurableEquiv.piCongrLeft β e
refine ⟨η.map E, inferInstance, fun a ↦ ?_⟩
rw [Kernel.map_apply _ E.measurable, heq]
exact (measurePreserving_piCongrLeft (fun i ↦ κ i a) e).map_eq
noncomputable def finiteKernelPi {α ι : Type*} [MeasurableSpace α] [Fintype ι]
{β : ι → Type*} [∀ i, MeasurableSpace (β i)]
(κ : ∀ i, Kernel α (β i)) [∀ i, IsFiniteKernel (κ i)] : Kernel α (∀ i, β i) :=
(exists_kernel_pi β κ).choose
instance {α ι : Type*} [MeasurableSpace α] [Fintype ι]
{β : ι → Type*} [∀ i, MeasurableSpace (β i)]
(κ : ∀ i, Kernel α (β i)) [∀ i, IsFiniteKernel (κ i)] :
IsFiniteKernel (finiteKernelPi κ) := (exists_kernel_pi β κ).choose_spec.1
lemma finiteKernelPi_apply {α ι : Type*} [MeasurableSpace α] [Fintype ι]
{β : ι → Type*} [∀ i, MeasurableSpace (β i)]
(κ : ∀ i, Kernel α (β i)) [∀ i, IsFiniteKernel (κ i)] (a : α) :
finiteKernelPi κ a = Measure.pi (fun i ↦ κ i a) :=
(exists_kernel_pi β κ).choose_spec.2 a
private lemma lintegral_fin_prod {n : ℕ} {β : Fin n → Type*}
[∀ i, MeasurableSpace (β i)] (μ : ∀ i, Measure (β i)) [∀ i, SigmaFinite (μ i)]
(f : ∀ i, β i → ℝ≥0∞) (hf : ∀ i, Measurable (f i)) :
(∫⁻ x, ∏ i, f i (x i) ∂Measure.pi μ) = ∏ i, ∫⁻ x, f i x ∂μ i := by
classical
induction n with
| zero => simp [Measure.pi_of_empty μ]
| succ n ih =>
have hp := (measurePreserving_piFinSuccAbove μ 0).symm
(MeasurableEquiv.piFinSuccAbove β 0)
rw [← hp.lintegral_comp (f := fun x ↦ ∏ i, f i (x i)) (Finset.measurable_prod _ fun i _ ↦
(hf i).comp (measurable_pi_apply i))]
have heq : (fun z ↦ ∏ i, f i ((MeasurableEquiv.piFinSuccAbove β 0).symm z i)) =
(fun z ↦ f 0 z.1 * ∏ j, f ((0 : Fin (n+1)).succAbove j) (z.2 j)) := by
funext z
rw [Fin.prod_univ_succAbove _ 0]
simp only [MeasurableEquiv.piFinSuccAbove_symm_apply, Fin.insertNthEquiv,
Equiv.coe_fn_mk, Fin.insertNth_apply_same, Fin.insertNth_apply_succAbove]
rw [heq]
have hm : Measurable (fun z : ∀ j : Fin n, β ((0 : Fin (n+1)).succAbove j) ↦
∏ j, f ((0 : Fin (n+1)).succAbove j) (z j)) :=
Finset.measurable_prod _ fun i _ ↦
(hf ((0 : Fin (n+1)).succAbove i)).comp (measurable_pi_apply i)
rw [lintegral_prod_mul (hf 0).aemeasurable hm.aemeasurable]
rw [ih _ _ (fun i ↦ hf ((0 : Fin (n+1)).succAbove i))]
exact (Fin.prod_univ_succAbove (fun i ↦ ∫⁻ x, f i x ∂μ i) 0).symm
lemma lintegral_finite_prod {ι : Type*} [Fintype ι] {β : ι → Type*}
[∀ i, MeasurableSpace (β i)] (μ : ∀ i, Measure (β i)) [∀ i, SigmaFinite (μ i)]
(f : ∀ i, β i → ℝ≥0∞) (hf : ∀ i, Measurable (f i)) :
(∫⁻ x, ∏ i, f i (x i) ∂Measure.pi μ) = ∏ i, ∫⁻ x, f i x ∂μ i := by
classical
let e := (Fintype.equivFin ι).symm
rw [← (measurePreserving_piCongrLeft μ e).lintegral_comp (f := fun x ↦ ∏ i, f i (x i))
(Finset.measurable_prod _ fun i _ ↦ (hf i).comp (measurable_pi_apply i))]
simp_rw [← e.prod_comp, MeasurableEquiv.coe_piCongrLeft,
Equiv.piCongrLeft_apply_apply]
exact lintegral_fin_prod _ _ (fun i ↦ hf (e i))
section CoordinateKernels
variable {ι : Type*} [Fintype ι] {X Z : ι → Type*}
[∀ i, MeasurableSpace (X i)] [∀ i, MeasurableSpace (Z i)]
(κ : ∀ i, Kernel (X i) (Z i)) [∀ i, IsFiniteKernel (κ i)]
def coordinateKernel : Kernel (∀ i, X i) (∀ i, Z i) :=
finiteKernelPi (fun i ↦ (κ i).comap (fun x ↦ x i) (measurable_pi_apply (i : ι)))
instance : IsFiniteKernel (coordinateKernel κ) := by
unfold coordinateKernel
infer_instance
lemma coordinateKernel_apply (x : ∀ i, X i) :
coordinateKernel κ x = Measure.pi (fun i ↦ κ i (x i)) := by
simp only [coordinateKernel, finiteKernelPi_apply, Kernel.comap_apply]
lemma pi_compProd (μ : ∀ i, Measure (X i)) [∀ i, IsFiniteMeasure (μ i)] :
Measure.map (fun p : (∀ i, X i) × (∀ i, Z i) ↦ fun i ↦ (p.1 i, p.2 i))
((Measure.pi μ).compProd (coordinateKernel κ)) =
Measure.pi (fun i ↦ (μ i).compProd (κ i)) := by
classical
have hm : Measurable (fun p : (∀ i, X i) × (∀ i, Z i) ↦ fun i ↦ (p.1 i, p.2 i)) := by
fun_prop
symm
apply Measure.pi_eq
intro s hs
rw [Measure.map_apply hm (MeasurableSet.univ_pi hs),
Measure.compProd_apply (hm (MeasurableSet.univ_pi hs))]
have heq (x : ∀ i, X i) : (Prod.mk x ⁻¹'
((fun p : (∀ i, X i) × (∀ i, Z i) ↦ fun i ↦ (p.1 i, p.2 i)) ⁻¹' Set.pi Set.univ s)) =
Set.pi Set.univ (fun i ↦ Prod.mk (x i) ⁻¹' s i) := by
ext z
simp
simp_rw [heq, coordinateKernel_apply, Measure.pi_pi]
rw [lintegral_finite_prod _ _ (fun i ↦ Kernel.measurable_kernel_prodMk_left (hs i))]
congr 1
funext i
exact (Measure.compProd_apply (hs i)).symm
lemma measurable_kernelPi_integral {f : (∀ i, X i) × (∀ i, Z i) → ℝ≥0∞}
(hf : Measurable f) :
Measurable (fun x ↦ ∫⁻ z, f (x,z) ∂Measure.pi (fun i ↦ κ i (x i))) := by
simpa only [coordinateKernel_apply] using
(hf.lintegral_kernel_prod_right' (κ := coordinateKernel κ))
lemma lintegral_pi_compProd (μ : ∀ i, Measure (X i)) [∀ i, IsFiniteMeasure (μ i)]
{f : (∀ i, X i × Z i) → ℝ≥0∞} (hf : Measurable f) :
(∫⁻ p, f p ∂Measure.pi (fun i ↦ (μ i).compProd (κ i))) =
∫⁻ x, ∫⁻ z, f (fun i ↦ (x i,z i)) ∂Measure.pi (fun i ↦ κ i (x i)) ∂Measure.pi μ := by
have hm : Measurable (fun p : (∀ i, X i) × (∀ i, Z i) ↦ fun i ↦ (p.1 i, p.2 i)) := by
fun_prop
rw [← pi_compProd κ μ, lintegral_map hf hm]
rw [Measure.lintegral_compProd (f := fun p : (∀ i, X i) × (∀ i, Z i) ↦ f (fun i ↦ (p.1 i, p.2 i)))
(hf.comp hm)]
simp only [coordinateKernel_apply]
end CoordinateKernels
end HardSphere.Comparison
noncomputable section
open MeasureTheory Function Set Filter
open scoped ENNReal BigOperators
end
end
end
end
end OAI
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