| import Mathlib |
|
|
| 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 |
|
|
