| import OAI.Geometry.PinnedDistances.VarianceRates |
|
|
| namespace OAI |
|
|
| universe u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12 |
| universe u13 u14 |
|
|
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology BigOperators BoundedContinuousFunction |
| namespace WeakPinned.Bounded |
| |
| variable {E : Type u1} {X : Type u2} {J : Type u3} [MeasurableSpace E] [MeasurableSpace X] |
| [TopologicalSpace E] [TopologicalSpace X] [BorelSpace E] [BorelSpace X] |
| [SecondCountableTopology E] [SecondCountableTopology X] |
| [HasOuterApproxClosed E] [HasOuterApproxClosed X] [Fintype J] |
| |
| omit [SecondCountableTopology E] in |
| lemma weak_product_limit |
| (Rn : ℕ → ProbabilityMeasure (E × (J → X))) (R : ProbabilityMeasure (E × (J → X))) |
| (Qn : ℕ → ProbabilityMeasure E) (Q : ProbabilityMeasure E) |
| (αn : ℕ → J → ProbabilityMeasure X) (β : J → ProbabilityMeasure X) |
| (hR : Tendsto Rn atTop (𝓝 R)) (hQ : Tendsto Qn atTop (𝓝 Q)) |
| (hα : ∀ j,Tendsto (fun n => αn n j) atTop (𝓝 (β j))) |
| (herr : ∀ (f : E →ᵇ ℝ) (g : J → X →ᵇ ℝ), |
| Tendsto (fun n => (∫ p,f p.1*(∏ j,g j (p.2 j)) ∂ (Rn n : Measure (E × (J → X))))- |
| (∫ e,f e ∂ (Qn n : Measure E))*(∏ j,∫ x,g j x ∂ (αn n j : Measure X))) atTop (𝓝 0)) : |
| (R : Measure (E × (J → X)))=(Q : Measure E).prod (Measure.pi (fun j => (β j : Measure X))) := by |
| apply Measure.eq_prod_of_integral_mul_prod_boundedContinuousFunction |
| intro f g |
| let F : E × (J → X) →ᵇ ℝ := |
| f.compContinuous ⟨Prod.fst,continuous_fst⟩ * |
| ∏ j,(g j).compContinuous ⟨fun p : E × (J → X) => p.2 j,by fun_prop⟩ |
| have hRf := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hR F |
| have hQf := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hQ f |
| have hβg (j) := ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp (hα j) (g j) |
| have ht := hRf.sub (hQf.mul (tendsto_finsetProd Finset.univ (fun j _ => hβg j))) |
| have hF (p : E × (J → X)) : F p=f p.1 * ∏ j,g j (p.2 j) := by simp [F] |
| simp only [hF] at ht |
| have he := tendsto_nhds_unique ht (herr f g) |
| rw [integral_fintype_prod_eq_prod] |
| exact sub_eq_zero.mp he |
|
|
| end WeakPinned.Bounded |
| |
| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology BigOperators BoundedContinuousFunction |
| namespace WeakPinned.Bounded |
| |
| lemma weak_pair_product_limit {X : Type u4} {Y : Type u5} [MeasurableSpace X] [MeasurableSpace Y] |
| [TopologicalSpace X] [TopologicalSpace Y] [BorelSpace X] [BorelSpace Y] |
| [HasOuterApproxClosed X] [HasOuterApproxClosed Y] |
| [SecondCountableTopology X] [SecondCountableTopology Y] |
| (Rn : ℕ → ProbabilityMeasure (X × Y)) (R : ProbabilityMeasure (X × Y)) |
| (μn : ℕ → ProbabilityMeasure X) (μ : ProbabilityMeasure X) |
| (νn : ℕ → ProbabilityMeasure Y) (ν : ProbabilityMeasure Y) |
| (hR : Tendsto Rn atTop (𝓝 R)) (hμ : Tendsto μn atTop (𝓝 μ)) (hν : Tendsto νn atTop (𝓝 ν)) |
| (herr : ∀ (f : X →ᵇ ℝ) (g : Y →ᵇ ℝ), |
| Tendsto (fun n => (∫ p,f p.1*g p.2 ∂ (Rn n : Measure (X × Y)))- |
| (∫ x,f x ∂ (μn n : Measure X))*(∫ y,g y ∂ (νn n : Measure Y))) atTop (𝓝 0)) : |
| (R : Measure (X × Y))=(μ : Measure X).prod (ν : Measure Y) := by |
| apply Measure.eq_prod_of_integral_mul_boundedContinuousFunction |
| intro f g |
| let F : X × Y →ᵇ ℝ := f.compContinuous ⟨Prod.fst,continuous_fst⟩ * |
| g.compContinuous ⟨Prod.snd,continuous_snd⟩ |
| have ht := (ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hR F).sub |
| ((ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hμ f).mul |
| (ProbabilityMeasure.tendsto_iff_forall_integral_tendsto.mp hν g)) |
| exact sub_eq_zero.mp (tendsto_nhds_unique ht (herr f g)) |
|
|
| lemma pair_of_pi_map {T : Type u6} {E : Type u7} {X : Type u8} [MeasurableSpace T] [MeasurableSpace E] [MeasurableSpace X] |
| (R : Measure T) (Q : Measure E) (μ : Measure X) [SFinite Q] [IsFiniteMeasure μ] |
| (e : T → E) (z w : T → X) (he : Measurable e) (hz : Measurable z) (hw : Measurable w) |
| (hm : R.map (fun r => (e r,![z r,w r]))=Q.prod (Measure.pi (fun _ : Fin 2 => μ))) : |
| R.map (fun r => (e r,(z r,w r)))=Q.prod (μ.prod μ) := by |
| have hp : Measurable (fun r => (e r,![z r,w r])) := by |
| apply he.prodMk |
| apply Measurable.of_eval |
| intro j |
| fin_cases j |
| · exact hz |
| · exact hw |
| let hme := MeasurableEquiv.finTwoArrow (α:=X) |
| have hm2 : Measurable (Prod.map (id : E → E) hme) := measurable_id.prodMap hme.measurable |
| have hh := congrArg (fun m : Measure (E × (Fin 2 → X)) => m.map (Prod.map id hme)) hm |
| rw [Measure.map_map hm2 hp,← Measure.map_prod_map _ _ measurable_id hme.measurable, |
| Measure.map_id,(measurePreserving_finTwoArrow μ).map_eq] at hh |
| exact hh |
|
|
| end WeakPinned.Bounded |
| |
| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology BigOperators BoundedContinuousFunction |
| namespace WeakPinned.Bounded |
| |
| variable {E : Type u9} {X : Type u10} {T : Type u11} [MeasurableSpace E] [MeasurableSpace X] [MeasurableSpace T] |
| [TopologicalSpace E] [TopologicalSpace X] [TopologicalSpace T] |
| [BorelSpace E] [BorelSpace X] [BorelSpace T] |
| [SecondCountableTopology E] [SecondCountableTopology X] |
| [HasOuterApproxClosed E] [HasOuterApproxClosed X] |
|
|
| lemma weak_double_product_limit (Rn : ℕ → ProbabilityMeasure T) (R : ProbabilityMeasure T) |
| (Qn : ℕ → ProbabilityMeasure E) (Q : ProbabilityMeasure E) |
| (αn : ℕ → ProbabilityMeasure X) (β : ProbabilityMeasure X) |
| (e : T → E) (z w : T → X) (he : Continuous e) (hz : Continuous z) (hw : Continuous w) |
| (hR : Tendsto Rn atTop (𝓝 R)) (hQ : Tendsto Qn atTop (𝓝 Q)) (hα : Tendsto αn atTop (𝓝 β)) |
| (herr : ∀ (f : E →ᵇ ℝ) (g h : X →ᵇ ℝ), |
| Tendsto (fun n => (∫ r,f (e r)*g (z r)*h (w r) ∂ (Rn n : Measure T))- |
| (∫ q,f q ∂ (Qn n : Measure E))*(∫ x,g x ∂ (αn n : Measure X))* |
| (∫ x,h x ∂ (αn n : Measure X))) atTop (𝓝 0)) : |
| (R:Measure T).map (fun r => (e r,(z r,w r)))=(Q:Measure E).prod ((β:Measure X).prod β) := by |
| let k : T → E × (Fin 2 → X) := fun r => (e r,![z r,w r]) |
| have hk : Continuous k := by |
| apply he.prodMk |
| apply continuous_pi |
| intro i |
| fin_cases i |
| · exact hz |
| · exact hw |
| have hRm := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous Rn R hR hk |
| have hh := weak_product_limit (fun n => (Rn n).map k) |
| (R.map k) Qn Q (fun n _ => αn n) (fun _ : Fin 2 => β) hRm hQ (fun _ => hα) ?_ |
| · exact pair_of_pi_map (R:Measure T) (Q:Measure E) (β:Measure X) e z w he.measurable hz.measurable hw.measurable hh |
| · intro f g |
| have hl := herr f (g 0) (g 1) |
| convert hl using 1 |
| funext n |
| rw [show ((Rn n).map k : Measure (E × (Fin 2 → X)))=(Rn n:Measure T).map k from rfl] |
| rw [integral_map_of_stronglyMeasurable hk.measurable (by fun_prop)] |
| simp only [Fin.prod_univ_two,k,Matrix.cons_val_zero,Matrix.cons_val_one,Matrix.cons_val_fin_one,mul_assoc] |
| |
| end WeakPinned.Bounded |
| |
| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology ENNReal NNReal BoundedContinuousFunction |
| namespace WeakPinned.Bounded |
| |
| variable {X : Type u12} [MeasurableSpace X] [TopologicalSpace X] [BorelSpace X] |
| [TopologicalSpace.PseudoMetrizableSpace X] |
|
|
| lemma measure_le_of_continuous_lintegral (μ ν : Measure X) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| (h : ∀ f : X →ᵇ ℝ≥0,(∫⁻ x,f x ∂ μ)≤∫⁻ x,f x ∂ ν) : μ≤ν := by |
| have hclosed (K : Set X) (hK : IsClosed K) : μ K≤ν K := by |
| exact le_of_tendsto_of_tendsto |
| (HasOuterApproxClosed.tendsto_lintegral_apprSeq hK μ) |
| (HasOuterApproxClosed.tendsto_lintegral_apprSeq hK ν) |
| (Eventually.of_forall (fun n => h (hK.apprSeq n))) |
| apply Measure.le_iff.mpr |
| intro s hs |
| rw [hs.measure_eq_iSup_isClosed_of_ne_top (measure_ne_top μ s)] |
| exact iSup_le (fun K => iSup_le (fun hKs => iSup_le (fun hK => (hclosed K hK).trans (measure_mono hKs)))) |
|
|
| lemma weak_domination (μn νn : ℕ → ProbabilityMeasure X) (μ ν : ProbabilityMeasure X) |
| (hμ : Tendsto μn atTop (𝓝 μ)) (hν : Tendsto νn atTop (𝓝 ν)) (c : ℝ≥0) |
| (hle : ∀ n,(μn n : Measure X)≤(c:ℝ≥0∞) • (νn n : Measure X)) : |
| (μ : Measure X)≤(c:ℝ≥0∞) • (ν : Measure X) := by |
| have : IsFiniteMeasure ((c:ℝ≥0∞) • (ν:Measure X)) := (ν:Measure X).smul_finite ENNReal.coe_ne_top |
| apply measure_le_of_continuous_lintegral |
| intro f |
| have hμf := ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto.mp hμ f |
| have hνf := ProbabilityMeasure.tendsto_iff_forall_lintegral_tendsto.mp hν f |
| rw [lintegral_smul_measure] |
| apply le_of_tendsto_of_tendsto hμf |
| (ENNReal.Tendsto.const_mul hνf (Or.inr ENNReal.coe_ne_top)) |
| filter_upwards [] with n |
| simpa only [lintegral_smul_measure,smul_eq_mul] using lintegral_mono' (hle n) (le_refl (fun x => (f x:ℝ≥0∞))) |
| |
| end WeakPinned.Bounded |
| |
| end |
| end |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped Topology NNReal ENNReal |
| namespace WeakPinned.Bounded |
| variable {X : Type u13} {Y : Type u14} [MeasurableSpace X] [MeasurableSpace Y] |
| [TopologicalSpace X] [TopologicalSpace Y] [BorelSpace X] [BorelSpace Y] |
| [HasOuterApproxClosed Y] |
| |
| lemma weak_map_eq (αn : ℕ → ProbabilityMeasure X) (α : ProbabilityMeasure X) |
| (βn : ℕ → ProbabilityMeasure Y) (β : ProbabilityMeasure Y) |
| (f : X → Y) (hf : Continuous f) (hα : Tendsto αn atTop (𝓝 α)) |
| (hβ : Tendsto βn atTop (𝓝 β)) (hm : ∀ n,(αn n:Measure X).map f=(βn n:Measure Y)) : |
| (α:Measure X).map f=(β:Measure Y) := by |
| have he (n) : (αn n).map f=βn n := ProbabilityMeasure.toMeasure_injective (hm n) |
| have hl := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous αn α hα hf |
| simp only [he] at hl |
| exact congrArg ProbabilityMeasure.toMeasure (tendsto_nhds_unique hl hβ) |
| |
| omit [HasOuterApproxClosed Y] in |
| lemma weak_map_domination [TopologicalSpace.PseudoMetrizableSpace Y] |
| (αn : ℕ → ProbabilityMeasure X) (α : ProbabilityMeasure X) |
| (βn : ℕ → ProbabilityMeasure Y) (β : ProbabilityMeasure Y) |
| (f : X → Y) (hf : Continuous f) (hα : Tendsto αn atTop (𝓝 α)) |
| (hβ : Tendsto βn atTop (𝓝 β)) (c : ℝ≥0) |
| (hm : ∀ n,(αn n:Measure X).map f≤(c:ℝ≥0∞) • (βn n:Measure Y)) : |
| (α:Measure X).map f≪(β:Measure Y) := by |
| have hl := weak_domination (fun n => (αn n).map f) |
| βn (α.map f) β |
| (ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous αn α hα hf) hβ c hm |
| exact Measure.absolutelyContinuous_of_le_smul hl |
| |
| end WeakPinned.Bounded |
| |
| end |
| end |
| |
| end OAI |
| |
