| import OAI.Geometry.PinnedDistances.ProductLimits |
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| namespace OAI |
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| universe u1 u2 u3 u4 u5 u6 |
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| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped BigOperators Topology BoundedContinuousFunction NNReal ENNReal |
| namespace WeakPinned.Bounded |
| local instance : MeasurableSpace Sphere := borel Sphere |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
| variable {V : ℕ → Type u1} {I : ℕ → Type u2} [∀ n,Fintype (V n)] [∀ n,DecidableEq (V n)] [∀ n,Fintype (I n)] |
| variable (G : ∀ n,Core.Fibers (V n) (I n)) (U₁ U₂ : ∀ n,V n → Sphere) |
| (hG : ∀ n,0<(G n).edgeCount) (hB : ∀ n e,((G n).targets e).Nonempty) |
| (hn : ∀ n,2≤Fintype.card (V n)) |
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|
| lemma expanded_marginal_limit (π : Sphere × Sphere → Sphere) (hπ : Continuous π) |
| (β : ProbabilityMeasure Sphere) (Q : ProbabilityMeasure (Quad Sphere)) |
| (R : ProbabilityMeasure (Expanded Sphere)) |
| (hα : Tendsto (fun n => uniformProbability (fun x => π (U₁ n x,U₂ n x)) (by have:=hn n;omega)) atTop (𝓝 β)) |
| (hQ : Tendsto (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) atTop (𝓝 Q)) |
| (hR : Tendsto (fun n => expandedProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) atTop (𝓝 R)) |
| (hd : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep (G n).fiberWeight (G n).targets |
| (fun x => f (π (U₁ n x,U₂ n x)))) atTop (𝓝 0)) : |
| (R:Measure (Expanded Sphere)).map (fun r => (r.1,(π r.2.1,π r.2.2)))= |
| (Q:Measure (Quad Sphere)).prod ((β:Measure Sphere).prod β) := by |
| apply weak_double_product_limit (fun n => expandedProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) R |
| (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) Q |
| (fun n => uniformProbability (fun x => π (U₁ n x,U₂ n x)) (by have:=hn n;omega)) β |
| Prod.fst (fun r => π r.2.1) (fun r => π r.2.2) continuous_fst |
| (hπ.comp (continuous_fst.comp continuous_snd)) (hπ.comp (continuous_snd.comp continuous_snd)) hR hQ hα |
| exact expanded_test_rate G U₁ U₂ hG hB hn π hd |
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|
| lemma incoming_limit (T : ℕ → ℝ) (hT : ∀ n,0<T n) |
| (β : ProbabilityMeasure Sphere) (Q : ProbabilityMeasure (Quad Sphere)) |
| (hα : Tendsto (fun n => uniformProbability (U₁ n) (by have:=hn n;omega)) atTop (𝓝 β)) |
| (hQ : Tendsto (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) atTop (𝓝 Q)) |
| (hd : ∀ f : C(Sphere,ℝ),Tendsto (fun n => testDiscrep ((G n).retainedWeight (T n)) |
| ((G n).retainedSet (T n)) (fun x => f (U₁ n x))) atTop (𝓝 0)) |
| (ho : Tendsto (fun n => (G n).omittedMass (T n)) atTop (𝓝 0)) : |
| (Q:Measure (Quad Sphere)).map (fun q => (q.1,q.2.2))= |
| (β:Measure Sphere).prod ((Q:Measure (Quad Sphere)).map (fun q => q.2.2)) := by |
| let Qn := fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n) |
| have hc : Continuous (fun q : Quad Sphere => (q.1,q.2.2)) := by fun_prop |
| have ht : Continuous (fun q : Quad Sphere => q.2.2) := by fun_prop |
| apply weak_pair_product_limit (fun n => (Qn n).map (fun q => (q.1,q.2.2))) |
| (Q.map (fun q => (q.1,q.2.2))) |
| (fun n => uniformProbability (U₁ n) (by have:=hn n;omega)) β |
| (fun n => (Qn n).map (fun q => q.2.2)) (Q.map (fun q => q.2.2)) |
| (ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous Qn Q hQ hc) hα |
| (ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous Qn Q hQ ht) |
| intro f g |
| have hh := incoming_law_test_rate G U₁ U₂ hG hB hn T hT hd ho f g |
| convert hh using 1 |
| funext n |
| change (∫ q,f q.1*g q.2 ∂ (Qn n:Measure (Quad Sphere)).map (fun q => (q.1,q.2.2)))- |
| (∫ z,f z ∂ uniformLaw (U₁ n))*(∫ p,g p ∂ (Qn n:Measure (Quad Sphere)).map (fun q => q.2.2))=_ |
| rw [integral_map_of_stronglyMeasurable hc.measurable (by fun_prop), |
| integral_map_of_stronglyMeasurable ht.measurable g.continuous.stronglyMeasurable] |
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| lemma edge_ac_limit (β₁ β₂ : ProbabilityMeasure Sphere) (Q : ProbabilityMeasure (Quad Sphere)) |
| (hα₁ : Tendsto (fun n => uniformProbability (U₁ n) (by have:=hn n;omega)) atTop (𝓝 β₁)) |
| (hα₂ : Tendsto (fun n => uniformProbability (U₂ n) (by have:=hn n;omega)) atTop (𝓝 β₂)) |
| (hQ : Tendsto (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) atTop (𝓝 Q)) |
| (M : ℝ) (hM : 0≤M) (hdom : ∀ n x y,(G n).edgeLaw x y≤M*Variance.pairProb Finset.univ x y) : |
| (Q:Measure (Quad Sphere)).map (fun q => (q.1,q.2.2.1))≪(β₁:Measure Sphere).prod β₁ ∧ |
| (Q:Measure (Quad Sphere)).map (fun q => (q.2.1,q.2.2.2))≪(β₂:Measure Sphere).prod β₂ := by |
| let c : ℝ≥0 := ⟨2*M,by positivity⟩ |
| have hc : (c:ℝ≥0∞)=ENNReal.ofReal (2*M) := by |
| rw [ENNReal.ofReal_eq_coe_nnreal (by positivity)] |
| rfl |
| constructor |
| · apply weak_map_domination |
| (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) Q |
| (fun n => (uniformProbability (U₁ n) (by have:=hn n;omega)).prod |
| (uniformProbability (U₁ n) (by have:=hn n;omega))) (β₁.prod β₁) |
| (fun q => (q.1,q.2.2.1)) (by fun_prop) hQ |
| (ProbabilityMeasure.continuous_prod.tendsto (β₁,β₁) |>.comp (hα₁.prodMk_nhds hα₁)) c |
| intro n |
| rw [hc] |
| exact edgeProbability_dom₁ (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hn n) M hM (hdom n) |
| · apply weak_map_domination |
| (fun n => edgeProbability (G n) (U₁ n) (U₂ n) (hG n) (hB n)) Q |
| (fun n => (uniformProbability (U₂ n) (by have:=hn n;omega)).prod |
| (uniformProbability (U₂ n) (by have:=hn n;omega))) (β₂.prod β₂) |
| (fun q => (q.2.1,q.2.2.2)) (by fun_prop) hQ |
| (ProbabilityMeasure.continuous_prod.tendsto (β₂,β₂) |>.comp (hα₂.prodMk_nhds hα₂)) c |
| intro n |
| rw [hc] |
| exact edgeProbability_dom₂ (G n) (U₁ n) (U₂ n) (hG n) (hB n) (hn n) M hM (hdom n) |
|
|
| end WeakPinned.Bounded |
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| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped Topology |
| namespace WeakPinned.Bounded |
|
|
| local instance : MeasurableSpace Sphere := borel Sphere |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
|
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| lemma quadCoe_range (q : Quad Sphere) |
| (h₁ : (q.1,q.2.2.1)∈range pairCoe) (h₂ : (q.2.1,q.2.2.2)∈range pairCoe) : |
| q∈range quadCoe := by |
| obtain ⟨⟨a,c⟩,ha⟩ := h₁ |
| obtain ⟨⟨b,d⟩,hb⟩ := h₂ |
| exact ⟨(a,b,c,d),by ext <;> simp_all [pairCoe,quadCoe,Prod.mk.injEq]⟩ |
|
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| lemma expandedCoe_range (r : Expanded Sphere) |
| (h₁ : (r.1,(r.2.1.1,r.2.2.1))∈range (Prod.map quadCoe pairCoe)) |
| (h₂ : (r.1,(r.2.1.2,r.2.2.2))∈range (Prod.map quadCoe pairCoe)) : |
| r∈range expandedCoe := by |
| obtain ⟨⟨q,z,w⟩,hz⟩ := h₁ |
| obtain ⟨⟨q',v,t⟩,hv⟩ := h₂ |
| refine ⟨(q,((z,v),(w,t))),?_⟩ |
| have he : quadCoe q=r.1 := congrArg Prod.fst hz |
| have hz' := congrArg Prod.snd hz |
| have hv' := congrArg Prod.snd hv |
| apply Prod.ext he |
| dsimp [expandedCoe] |
| ext <;> simp_all [pairCoe,Prod.mk.injEq] |
| |
| lemma map_transport_ac {A : Type u3} {B : Type u4} {X : Type u5} {Y : Type u6} [MeasurableSpace A] [MeasurableSpace B] |
| [MeasurableSpace X] [MeasurableSpace Y] (j : A → X) (k : B → Y) |
| (hj : Measurable j) (hk : MeasurableEmbedding k) |
| (f : A → B) (g : X → Y) (hf : Measurable f) (hg : Measurable g) |
| (μ : Measure A) (ν : Measure B) (hcomm : k ∘ f=g ∘ j) |
| (hm : (μ.map j).map g≪ν.map k) : μ.map f≪ν := by |
| apply map_pull_ac k hk |
| rw [Measure.map_map hk.measurable hf,hcomm,← Measure.map_map hg hj] |
| exact hm |
| |
| lemma sphere_quad_finite (μ ν : Measure ℂ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (Q : Measure (Quad Sphere)) |
| (h₁ : Q.map (fun q => (q.1,q.2.2.1))≪(μ.map OnePoint.some).prod (μ.map OnePoint.some)) |
| (h₂ : Q.map (fun q => (q.2.1,q.2.2.2))≪(ν.map OnePoint.some).prod (ν.map OnePoint.some)) : |
| ∀ᵐ q ∂ Q,q∈range quadCoe := by |
| have hpμ := map_full_range pairCoe pairCoe_open.measurableEmbedding (μ.prod μ) |
| have hpν := map_full_range pairCoe pairCoe_open.measurableEmbedding (ν.prod ν) |
| rw [← Measure.map_prod_map _ _ measurableEmbedding_sphere_coe.measurable |
| measurableEmbedding_sphere_coe.measurable] at hpμ hpν |
| have hq₁ := (ae_map_iff (by fun_prop) pairCoe_open.measurableEmbedding.measurableSet_range).mp |
| (h₁.ae_le hpμ) |
| have hq₂ := (ae_map_iff (by fun_prop) pairCoe_open.measurableEmbedding.measurableSet_range).mp |
| (h₂.ae_le hpν) |
| exact (hq₁.and hq₂).mono (fun q h => quadCoe_range q h.1 h.2) |
| |
| lemma sphere_expanded_finite (μ ν : Measure ℂ) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (Q : Measure (Quad ℂ)) (R : Measure (Expanded Sphere)) [IsProbabilityMeasure Q] |
| (h₁ : R.map (fun r => (r.1,(r.2.1.1,r.2.2.1)))= |
| (Q.map quadCoe).prod ((μ.map OnePoint.some).prod (μ.map OnePoint.some))) |
| (h₂ : R.map (fun r => (r.1,(r.2.1.2,r.2.2.2)))= |
| (Q.map quadCoe).prod ((ν.map OnePoint.some).prod (ν.map OnePoint.some))) : |
| ∀ᵐ r ∂ R,r∈range expandedCoe := by |
| have hj := quadCoe_open.measurableEmbedding.prodMap pairCoe_open.measurableEmbedding |
| have hpμ := map_full_range _ hj (Q.prod (μ.prod μ)) |
| have hpν := map_full_range _ hj (Q.prod (ν.prod ν)) |
| rw [← Measure.map_prod_map _ _ quadCoe_open.measurableEmbedding.measurable |
| pairCoe_open.measurableEmbedding.measurable, |
| ← Measure.map_prod_map _ _ measurableEmbedding_sphere_coe.measurable |
| measurableEmbedding_sphere_coe.measurable,← h₁] at hpμ |
| rw [← Measure.map_prod_map _ _ quadCoe_open.measurableEmbedding.measurable |
| pairCoe_open.measurableEmbedding.measurable, |
| ← Measure.map_prod_map _ _ measurableEmbedding_sphere_coe.measurable |
| measurableEmbedding_sphere_coe.measurable,← h₂] at hpν |
| have hq₁ := (ae_map_iff (by fun_prop) hj.measurableSet_range).mp hpμ |
| have hq₂ := (ae_map_iff (by fun_prop) hj.measurableSet_range).mp hpν |
| exact (hq₁.and hq₂).mono (fun r h => expandedCoe_range r h.1 h.2) |
| |
| end WeakPinned.Bounded |
| |
| end |
| end |
| |
| end OAI |
| |
