| import Mathlib |
| import OAI.Analysis.HardSphere.Comparison.AffinityNormalization |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Comparison |
| variable {α : Type*} [MeasurableSpace α] |
| lemma affinity_withDensity_real (μ : Measure α) [IsFiniteMeasure μ] |
| (p q : α→ℝ) (hp : Integrable p μ) (hq : Integrable q μ) |
| (hp0 : ∀ᵐx ∂μ,0≤p x) (hq0 : ∀ᵐx ∂μ,0≤q x) : |
| measureAffinity (μ.withDensity fun x ↦ ENNReal.ofReal (p x)) |
| (μ.withDensity fun x ↦ ENNReal.ofReal (q x))=affinity μ p q := by |
| let := isFiniteMeasure_withDensity_ofReal hp.2 |
| let := isFiniteMeasure_withDensity_ofReal hq.2 |
| rw [measureAffinity_eq_dominated _ _ μ (withDensity_absolutelyContinuous _ _) (withDensity_absolutelyContinuous _ _)] |
| unfold affinity |
| apply integral_congr_ae |
| filter_upwards [Measure.rnDeriv_withDensity₀ μ hp.aestronglyMeasurable.aemeasurable.ennreal_ofReal, |
| Measure.rnDeriv_withDensity₀ μ hq.aestronglyMeasurable.aemeasurable.ennreal_ofReal,hp0,hq0] with x hx hy hp' hq' |
| rw [hx,hy,ENNReal.toReal_ofReal hp',ENNReal.toReal_ofReal hq'] |
|
|
| variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (μ : Measure X) (ν : Measure Y) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| lemma swapLast_preserving : |
| MeasurePreserving (fun p : (Y×X)×X ↦ ((p.1.1,p.2),p.1.2)) |
| ((ν.prod μ).prod μ) ((ν.prod μ).prod μ) := by |
| have h := (measurePreserving_prodAssoc ν μ μ).symm.comp |
| (((MeasurePreserving.id ν).prod (Measure.measurePreserving_swap (μ:=μ) (ν:=μ))).comp |
| (measurePreserving_prodAssoc ν μ μ)) |
| exact h |
|
|
| lemma full_affinity_square (f : Y×X→ℝ) (hf : MemLp f 2 (ν.prod μ)) |
| (hf0 : ∀p,0≤f p) (V : ℝ) (hV : 0≤V) : |
| measureAffinity |
| (((ν.prod μ).prod μ).withDensity fun p ↦ ENNReal.ofReal (V*f p.1^2)) |
| (((ν.prod μ).prod μ).withDensity fun p ↦ ENNReal.ofReal (V*f (p.1.1,p.2)^2))= |
| V*∫Y,(∫x,f (Y,x) ∂μ)^2 ∂ν := by |
| have h₁ := hf.comp_fst μ |
| have h₂ : MemLp (fun p : (Y×X)×X ↦ f (p.1.1,p.2)) 2 ((ν.prod μ).prod μ) := |
| h₁.comp_measurePreserving (swapLast_preserving μ ν) |
| have hi₁ : Integrable (fun p : (Y×X)×X ↦ V*f p.1^2) ((ν.prod μ).prod μ) := |
| ((memLp_two_iff_integrable_sq h₁.aestronglyMeasurable).1 h₁).const_mul V |
| have hi₂ : Integrable (fun p : (Y×X)×X ↦ V*f (p.1.1,p.2)^2) ((ν.prod μ).prod μ) := |
| ((memLp_two_iff_integrable_sq h₂.aestronglyMeasurable).1 h₂).const_mul V |
| rw [affinity_withDensity_real _ _ _ hi₁ hi₂ |
| (Eventually.of_forall fun _ ↦ mul_nonneg hV (sq_nonneg _)) |
| (Eventually.of_forall fun _ ↦ mul_nonneg hV (sq_nonneg _))] |
| unfold affinity |
| have he : (fun p : (Y×X)×X ↦ Real.sqrt (V*f p.1^2)*Real.sqrt (V*f (p.1.1,p.2)^2))= |
| (fun p ↦ V*(f p.1*f (p.1.1,p.2))) := by |
| funext p |
| rw [Real.sqrt_mul hV,Real.sqrt_mul hV,Real.sqrt_sq (hf0 _),Real.sqrt_sq (hf0 _)] |
| have hh := Real.sq_sqrt hV |
| calc |
| _ = (Real.sqrt V)^2*(f p.1*f (p.1.1,p.2)) := by ring |
| _ = _ := by rw [hh] |
| rw [he,integral_const_mul] |
| congr 1 |
| have hprod := h₁.integrable_mul h₂ |
| have hA := (measurePreserving_prodAssoc ν μ μ).symm.integral_comp |
| (MeasurableEquiv.prodAssoc (α:=Y) (β:=X) (γ:=X)).symm.measurableEmbedding |
| (fun p : (Y×X)×X ↦ f p.1*f (p.1.1,p.2)) |
| rw [←hA] |
| rw [integral_prod] |
| · apply integral_congr_ae |
| exact Eventually.of_forall fun Y ↦ by |
| simpa [sq, MeasurableEquiv.prodAssoc] using (integral_prod_mul (μ:=μ) (ν:=μ) (fun x ↦ f (Y,x)) (fun x ↦ f (Y,x))) |
| · have hi : Integrable (fun p : (Y×X)×X ↦ f p.1*f (p.1.1,p.2)) |
| ((ν.prod (μ.prod μ)).map MeasurableEquiv.prodAssoc.symm) := by |
| rw [(measurePreserving_prodAssoc ν μ μ).symm.map_eq] |
| exact hprod |
| apply hi.comp_measurable |
| exact (measurePreserving_prodAssoc ν μ μ).symm.measurable |
|
|
| end HardSphere.Comparison |
|
|
| end |
|
|
| noncomputable section |
| open MeasureTheory |
| open scoped ENNReal |
| namespace HardSphere.Comparison |
| variable {X : Type*} [MeasurableSpace X] |
|
|
| lemma cell_density_normalization (μ : Measure X) (g : X→ℝ≥0∞) |
| (hg : Measurable g) (s : Set X) (N : ℕ) (V : ℝ) (hV : 0<V) : |
| ENNReal.ofReal ((N:ℝ)/V) • |
| ((μ.withDensity (fun x ↦ ENNReal.ofReal V*g x)).restrict s)= |
| N • ((μ.withDensity g).restrict s) := by |
| have he : μ.withDensity (fun x ↦ ENNReal.ofReal V*g x)= |
| ENNReal.ofReal V • μ.withDensity g := by |
| exact withDensity_smul _ hg |
| rw [he,Measure.restrict_smul,smul_smul] |
| have hc : ENNReal.ofReal ((N:ℝ)/V)*ENNReal.ofReal V=(N:ℝ≥0∞) := by |
| rw [←ENNReal.ofReal_mul (by positivity),div_mul_cancel₀ _ hV.ne'] |
| simp |
| rw [hc,Nat.cast_smul_eq_nsmul] |
| |
| end HardSphere.Comparison |
| |
| end |
| |
| noncomputable section |
| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Comparison |
| variable {α ι : Type*} [MeasurableSpace α] [Fintype ι] |
| |
| lemma sum_restrict_partition (μ : Measure α) (cell : α→ι) |
| (hcell : ∀i,MeasurableSet {x | cell x=i}) : |
| (∑i,μ.restrict {x | cell x=i})=μ := by |
| have hd : Pairwise (fun i j ↦ Disjoint {x | cell x=i} {x | cell x=j}) := by |
| intro i j hij |
| exact Set.disjoint_left.mpr (fun x hi hj ↦ hij (hi.symm.trans hj)) |
| have hu : (⋃i,{x | cell x=i})=univ := by ext x; simp |
| rw [←Measure.sum_fintype,←Measure.restrict_iUnion hd hcell,hu,Measure.restrict_univ] |
| |
| lemma affinity_partition_le (P Q : Measure α) [IsFiniteMeasure P] [IsFiniteMeasure Q] |
| (cell : α→ι) (hcell : ∀i,MeasurableSet {x | cell x=i}) : |
| (∑i,measureAffinity (P.restrict {x | cell x=i}) (Q.restrict {x | cell x=i}))≤ |
| measureAffinity P Q := by |
| have h := measureAffinity_finset_sum Finset.univ |
| (fun i ↦ P.restrict {x | cell x=i}) (fun i ↦ Q.restrict {x | cell x=i}) |
| simpa only [sum_restrict_partition P cell hcell, |
| sum_restrict_partition Q cell hcell] using h |
| |
| variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (μ : Measure X) (ν : Measure Y) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| |
| lemma cell_affinity_sum_le (f : Y×X→ℝ) (hf : MemLp f 2 (ν.prod μ)) |
| (hf0 : ∀p,0≤f p) (V : ℝ) (hV : 0≤V) (cell : X→ι) |
| (hcell : ∀i,MeasurableSet {x | cell x=i}) : |
| (∑ij : ι×ι,measureAffinity |
| ((((ν.prod μ).prod μ).withDensity fun p ↦ ENNReal.ofReal (V*f p.1^2)).restrict |
| {p | (cell p.1.2,cell p.2)=ij}) |
| ((((ν.prod μ).prod μ).withDensity fun p ↦ ENNReal.ofReal (V*f (p.1.1,p.2)^2)).restrict |
| {p | (cell p.1.2,cell p.2)=ij}))≤ |
| V*∫Y,(∫x,f (Y,x) ∂μ)^2 ∂ν := by |
| have h₁ := hf.comp_fst μ |
| have h₂ : MemLp (fun p : (Y×X)×X ↦ f (p.1.1,p.2)) 2 ((ν.prod μ).prod μ) := |
| h₁.comp_measurePreserving (swapLast_preserving μ ν) |
| have hi₁ : Integrable (fun p : (Y×X)×X ↦ V*f p.1^2) ((ν.prod μ).prod μ) := |
| ((memLp_two_iff_integrable_sq h₁.aestronglyMeasurable).1 h₁).const_mul V |
| have hi₂ : Integrable (fun p : (Y×X)×X ↦ V*f (p.1.1,p.2)^2) ((ν.prod μ).prod μ) := |
| ((memLp_two_iff_integrable_sq h₂.aestronglyMeasurable).1 h₂).const_mul V |
| let := isFiniteMeasure_withDensity_ofReal hi₁.2 |
| let := isFiniteMeasure_withDensity_ofReal hi₂.2 |
| rw [←full_affinity_square μ ν f hf hf0 V hV] |
| apply affinity_partition_le |
| intro ij |
| have he : {p : (Y×X)×X | (cell p.1.2,cell p.2)=ij}= |
| {p | cell p.1.2=ij.1}∩{p | cell p.2=ij.2} := by |
| ext p |
| exact Prod.ext_iff |
| rw [he] |
| exact ((hcell ij.1).preimage (measurable_snd.comp measurable_fst)).inter |
| ((hcell ij.2).preimage measurable_snd) |
| |
| end HardSphere.Comparison |
| |
| end |
| |
| end OAI |
| |
