| import OAI.Geometry.PinnedDistances.FiniteMixtures |
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| namespace OAI |
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| universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 |
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| section |
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped Topology |
| namespace WeakPinned.Bounded |
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| variable {X : Type u1} {Y : Type u2} [MeasurableSpace X] [MeasurableSpace Y] |
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| lemma chart_probability (j : X → Y) (hj : MeasurableEmbedding j) |
| (μ : Measure Y) [IsProbabilityMeasure μ] (hfin : ∀ᵐ y ∂ μ,y∈range j) : |
| IsProbabilityMeasure (μ.comap j) := by |
| constructor |
| rw [hj.comap_apply,image_univ] |
| exact measure_eq_measure_of_null_sdiff (subset_univ _) (by |
| convert ae_iff.mp hfin using 2 |
| ext y |
| simp) |>.trans measure_univ |
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| lemma chart_map_comap (j : X → Y) (hj : MeasurableEmbedding j) |
| (μ : Measure Y) (hfin : ∀ᵐ y ∂ μ,y∈range j) : (μ.comap j).map j=μ := by |
| rw [hj.map_comap] |
| exact Measure.restrict_eq_self_of_ae_mem hfin |
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| lemma chart_ae_iff (j : X → Y) (hj : MeasurableEmbedding j) (μ : Measure Y) |
| {s : Set X} : (∀ᵐ x ∂ μ.comap j,x∈s) ↔ μ (j '' sᶜ)=0 := by |
| rw [ae_iff,hj.comap_apply] |
| rfl |
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| variable [TopologicalSpace X] [TopologicalSpace Y] [BorelSpace X] [BorelSpace Y] |
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| lemma weak_chart_closed (j : X → Y) (hj : Topology.IsOpenEmbedding j) |
| [HasOuterApproxClosed Y] (α : ℕ → ProbabilityMeasure Y) (β : ProbabilityMeasure Y) |
| (ht : Tendsto α atTop (𝓝 β)) (s : Set X) (hs : IsClosed s) |
| (hα : ∀ n,(α n : Measure Y) (j '' sᶜ)=0) : |
| ∀ᵐ x ∂ (β : Measure Y).comap j,x∈s := by |
| rw [chart_ae_iff j hj.measurableEmbedding] |
| apply le_antisymm _ zero_le |
| apply (ProbabilityMeasure.le_liminf_measure_open_of_tendsto ht |
| (hj.isOpenMap _ hs.isOpen_compl)).trans |
| apply liminf_le_of_frequently_le' |
| exact (Eventually.of_forall (fun n => le_of_eq (hα n))).frequently |
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| end WeakPinned.Bounded |
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| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped Topology |
| namespace WeakPinned.Bounded |
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| lemma map_transport {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 hk.map_injective |
| rw [Measure.map_map hk.measurable hf,hcomm,← Measure.map_map hg hj] |
| exact hm |
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| lemma map_pull_ac {A : Type u7} {B : Type u8} [MeasurableSpace A] [MeasurableSpace B] |
| (j : A → B) (hj : MeasurableEmbedding j) (μ ν : Measure A) |
| (h : μ.map j ≪ ν.map j) : μ≪ν := by |
| intro s hs |
| have hν : ν.map j (j '' s)=0 := by |
| rw [hj.map_apply,hj.injective.preimage_image] |
| exact hs |
| have hμ := h hν |
| rwa [hj.map_apply,hj.injective.preimage_image] at hμ |
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| lemma map_full_range {A : Type u9} {B : Type u10} [MeasurableSpace A] [MeasurableSpace B] |
| (j : A → B) (hj : MeasurableEmbedding j) (μ : Measure A) : |
| ∀ᵐ x ∂ μ.map j,x∈range j := by |
| apply (ae_map_iff hj.measurable.aemeasurable hj.measurableSet_range).mpr |
| exact Eventually.of_forall (fun _ => mem_range_self _) |
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| abbrev pairCoe : ℂ × ℂ → Sphere × Sphere := Prod.map OnePoint.some OnePoint.some |
| abbrev quadCoe : Quad ℂ → Quad Sphere := Prod.map OnePoint.some |
| (Prod.map OnePoint.some pairCoe) |
| abbrev expandedCoe : Expanded ℂ → Expanded Sphere := Prod.map quadCoe (Prod.map pairCoe pairCoe) |
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| lemma pairCoe_open : Topology.IsOpenEmbedding pairCoe := |
| OnePoint.isOpenEmbedding_coe.prodMap OnePoint.isOpenEmbedding_coe |
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| lemma quadCoe_open : Topology.IsOpenEmbedding quadCoe := |
| OnePoint.isOpenEmbedding_coe.prodMap (OnePoint.isOpenEmbedding_coe.prodMap pairCoe_open) |
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| lemma expandedCoe_open : Topology.IsOpenEmbedding expandedCoe := |
| quadCoe_open.prodMap (pairCoe_open.prodMap pairCoe_open) |
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| end WeakPinned.Bounded |
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| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped BigOperators Topology ENNReal |
| namespace WeakPinned.Bounded |
| open Variance |
| variable {V : Type u11} {I : Type u12} [Fintype V] [DecidableEq V] [Fintype I] |
| variable (G : Core.Fibers V I) |
| local instance : MeasurableSpace Sphere := borel Sphere |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
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| omit [Fintype I] in |
| lemma graph_edgeWeight_nonneg (r : I × V) : |
| 0≤edgeWeight G.fiberWeight (fun e => prob (G.targets e)) r := |
| mul_nonneg (G.fiberWeight_nonneg _) (Core.Fibers.prob_nn _ _) |
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| omit [Fintype I] in |
| lemma graph_tripleWeight_nonneg (r : I × V × V × V) : |
| 0≤tripleWeight G.fiberWeight (fun e => prob (G.targets e)) r := |
| mul_nonneg (mul_nonneg (mul_nonneg (G.fiberWeight_nonneg _) (Core.Fibers.prob_nn _ _)) |
| (Core.Fibers.prob_nn _ _)) (Core.Fibers.prob_nn _ _) |
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| def edgeProbability (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) : ProbabilityMeasure (Quad Sphere) := |
| finiteProbability (edgeWeight G.fiberWeight (fun e => prob (G.targets e))) |
| (edgeValue G.source U₁ U₂) (graph_edgeWeight_nonneg G) |
| (sum_edgeWeight _ _ (G.sum_fiberWeight hG) (fun e => sum_prob _ (hB e))) |
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| def expandedProbability (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) : ProbabilityMeasure (Expanded Sphere) := |
| finiteProbability (tripleWeight G.fiberWeight (fun e => prob (G.targets e))) |
| (tripleValue G.source U₁ U₂) (graph_tripleWeight_nonneg G) |
| (sum_tripleWeight _ _ (G.sum_fiberWeight hG) (fun e => sum_prob _ (hB e))) |
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| lemma expandedProbability_edge (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) : |
| (expandedProbability G U₁ U₂ hG hB:Measure (Expanded Sphere)).map Prod.fst= |
| (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)) := by |
| change (finiteLaw _ _).map _=finiteLaw _ _ |
| rw [finiteLaw_map _ _ _ measurable_fst] |
| exact tripleLaw_edge G.fiberWeight (fun e => prob (G.targets e)) (edgeValue G.source U₁ U₂) G.fiberWeight_nonneg (fun e => Core.Fibers.prob_nn _) (fun e => sum_prob _ (hB e)) |
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| lemma edgeProbability_dom₁ (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) (hn : 2≤Fintype.card V) (M : ℝ) (hM : 0≤M) |
| (hdom : ∀ x y,G.edgeLaw x y≤M*pairProb Finset.univ x y) : |
| (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)).map (fun q => (q.1,q.2.2.1))≤ |
| ENNReal.ofReal (2*M) • (uniformLaw U₁).prod (uniformLaw U₁) := by |
| change (finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e))) |
| (edgeValue G.source U₁ U₂)).map (fun q => (q.1,q.2.2.1))≤_ |
| rw [finiteLaw_map _ _ (fun q : Quad Sphere => (q.1,q.2.2.1)) (by fun_prop)] |
| change finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e))) (fun r => (U₁ (G.source r.1),U₁ r.2))≤_ |
| rw [graph_edge_law G (fun x y => (U₁ x,U₁ y)) hG hB] |
| exact graph_domination G U₁ hn M hM hdom |
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| lemma edgeProbability_dom₂ (U₁ U₂ : V → Sphere) (hG : 0<G.edgeCount) |
| (hB : ∀ e,(G.targets e).Nonempty) (hn : 2≤Fintype.card V) (M : ℝ) (hM : 0≤M) |
| (hdom : ∀ x y,G.edgeLaw x y≤M*pairProb Finset.univ x y) : |
| (edgeProbability G U₁ U₂ hG hB:Measure (Quad Sphere)).map (fun q => (q.2.1,q.2.2.2))≤ |
| ENNReal.ofReal (2*M) • (uniformLaw U₂).prod (uniformLaw U₂) := by |
| change (finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e))) |
| (edgeValue G.source U₁ U₂)).map (fun q => (q.2.1,q.2.2.2))≤_ |
| rw [finiteLaw_map _ _ (fun q : Quad Sphere => (q.2.1,q.2.2.2)) (by fun_prop)] |
| change finiteLaw (edgeWeight G.fiberWeight (fun e => prob (G.targets e))) (fun r => (U₂ (G.source r.1),U₂ r.2))≤_ |
| rw [graph_edge_law G (fun x y => (U₂ x,U₂ y)) hG hB] |
| exact graph_domination G U₂ hn M hM hdom |
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| end WeakPinned.Bounded |
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| end |
| end |
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| end OAI |
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