| import Mathlib |
| import OAI.Analysis.HardSphere.Comparison.Affinity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.Comparison |
| variable {α : Type*} [MeasurableSpace α] |
| lemma measureAffinity_eq_dominated (P Q ξ : Measure α) |
| [IsFiniteMeasure P] [IsFiniteMeasure Q] [IsFiniteMeasure ξ] |
| (hP : P≪ξ) (hQ : Q≪ξ) : |
| measureAffinity P Q=affinity ξ (fun x ↦ (P.rnDeriv ξ x).toReal) |
| (fun x ↦ (Q.rnDeriv ξ x).toReal) := by |
| have hsum : P+Q≪ξ := by |
| intro s hs |
| simp only [Measure.add_apply,hP hs,hQ hs,add_zero] |
| have hp : P≪P+Q := Measure.AbsolutelyContinuous.rfl.add_right Q |
| have hq : Q≪P+Q := Measure.AbsolutelyContinuous.rfl.add_right' P |
| unfold measureAffinity affinity |
| rw [←integral_toReal_rnDeriv_mul hsum] |
| apply integral_congr_ae |
| filter_upwards [real_rnDeriv_chain (ξ:=ξ) hp,real_rnDeriv_chain (ξ:=ξ) hq] with x hx hy |
| rw [←hx,←hy,Real.sqrt_mul ENNReal.toReal_nonneg,Real.sqrt_mul ENNReal.toReal_nonneg] |
| have hsq := Real.sq_sqrt (show 0≤((P+Q).rnDeriv ξ x).toReal from ENNReal.toReal_nonneg) |
| nth_rw 1 [←hsq] |
| ring |
|
|
| lemma affinity_finset_sum {ι : Type*} (s : Finset ι) (μ : Measure α) |
| (p q : ι→α→ℝ) (hp : ∀i∈s,Integrable (p i) μ) (hq : ∀i∈s,Integrable (q i) μ) |
| (hp0 : ∀i x,0≤p i x) (hq0 : ∀i x,0≤q i x) : |
| (∑i∈s,affinity μ (p i) (q i))≤ |
| affinity μ (fun x ↦ ∑i∈s,p i x) (fun x ↦ ∑i∈s,q i x) := by |
| unfold affinity |
| rw [←integral_finsetSum s (fun i hi ↦ affinity_integrable (hp i hi) (hq i hi) |
| (Eventually.of_forall (hp0 i)) (Eventually.of_forall (hq0 i)))] |
| apply integral_mono |
| · exact integrable_finsetSum _ (fun i hi ↦ affinity_integrable (hp i hi) (hq i hi) |
| (Eventually.of_forall (hp0 i)) (Eventually.of_forall (hq0 i))) |
| · exact affinity_integrable (integrable_finsetSum _ hp) (integrable_finsetSum _ hq) |
| (Eventually.of_forall (fun x ↦ Finset.sum_nonneg (fun i _ ↦ hp0 i x))) |
| (Eventually.of_forall (fun x ↦ Finset.sum_nonneg (fun i _ ↦ hq0 i x))) |
| · intro x |
| exact Real.sum_sqrt_mul_sqrt_le s (fun i ↦ hp0 i x) (fun i ↦ hq0 i x) |
|
|
| lemma rnDeriv_real_mono (P Q ξ : Measure α) [IsFiniteMeasure P] [IsFiniteMeasure Q] |
| [IsFiniteMeasure ξ] (h : P≤Q) : |
| (fun x ↦ (P.rnDeriv ξ x).toReal)≤ᵐ[ξ](fun x ↦ (Q.rnDeriv ξ x).toReal) := by |
| have he := Measure.sub_add_cancel_of_le h |
| have hh := Measure.rnDeriv_add (Q-P) P ξ |
| rw [he] at hh |
| filter_upwards [hh,Measure.rnDeriv_lt_top P ξ,Measure.rnDeriv_lt_top (Q-P) ξ] with x hx hp hq |
| rw [hx,Pi.add_apply,ENNReal.toReal_add hq.ne hp.ne] |
| exact le_add_of_nonneg_left ENNReal.toReal_nonneg |
|
|
| lemma measureAffinity_mono (P Q P' Q' : Measure α) |
| [IsFiniteMeasure P] [IsFiniteMeasure Q] [IsFiniteMeasure P'] [IsFiniteMeasure Q'] |
| (hP : P≤P') (hQ : Q≤Q') : measureAffinity P Q ≤ measureAffinity P' Q' := by |
| have hP' : P'≪P'+Q' := Measure.AbsolutelyContinuous.rfl.add_right Q' |
| have hQ' : Q'≪P'+Q' := Measure.AbsolutelyContinuous.rfl.add_right' P' |
| have hp := (Measure.absolutelyContinuous_of_le hP).trans hP' |
| have hq := (Measure.absolutelyContinuous_of_le hQ).trans hQ' |
| rw [measureAffinity_eq_dominated P Q (P'+Q') hp hq] |
| unfold measureAffinity affinity |
| apply integral_mono_ae |
| · exact affinity_integrable Measure.integrable_toReal_rnDeriv Measure.integrable_toReal_rnDeriv |
| (Eventually.of_forall (fun _ ↦ ENNReal.toReal_nonneg)) |
| (Eventually.of_forall (fun _ ↦ ENNReal.toReal_nonneg)) |
| · exact affinity_integrable Measure.integrable_toReal_rnDeriv Measure.integrable_toReal_rnDeriv |
| (Eventually.of_forall (fun _ ↦ ENNReal.toReal_nonneg)) |
| (Eventually.of_forall (fun _ ↦ ENNReal.toReal_nonneg)) |
| · filter_upwards [rnDeriv_real_mono P P' (P'+Q') hP,rnDeriv_real_mono Q Q' (P'+Q') hQ] with x hx hy |
| exact mul_le_mul (Real.sqrt_le_sqrt hx) (Real.sqrt_le_sqrt hy) |
| (Real.sqrt_nonneg _) (Real.sqrt_nonneg _) |
|
|
| lemma rnDeriv_real_finset_sum {ι : Type*} (s : Finset ι) (P : ι→Measure α) |
| (ξ : Measure α) [∀i,IsFiniteMeasure (P i)] [IsFiniteMeasure ξ] : |
| (fun x ↦ ((∑i∈s,P i).rnDeriv ξ x).toReal)=ᵐ[ξ] |
| (fun x ↦ ∑i∈s,((P i).rnDeriv ξ x).toReal) := by |
| classical |
| induction s using Finset.induction_on with |
| | empty => |
| simp only [Finset.sum_empty] |
| filter_upwards [Measure.rnDeriv_zero ξ] with x hx |
| simp [hx] |
| | @insert i s hi hs => |
| simp only [Finset.sum_insert hi] |
| filter_upwards [Measure.rnDeriv_add (P i) (∑j∈s,P j) ξ,hs, |
| Measure.rnDeriv_lt_top (P i) ξ,Measure.rnDeriv_lt_top (∑j∈s,P j) ξ] with x hx hy ha hb |
| rw [hx,Pi.add_apply,ENNReal.toReal_add ha.ne hb.ne,hy] |
|
|
| lemma measureAffinity_finset_sum {ι : Type*} (s : Finset ι) (P Q : ι→Measure α) |
| [∀i,IsFiniteMeasure (P i)] [∀i,IsFiniteMeasure (Q i)] : |
| (∑i∈s,measureAffinity (P i) (Q i)) ≤ measureAffinity (∑i∈s,P i) (∑i∈s,Q i) := by |
| let ξ := (∑i∈s,P i)+(∑i∈s,Q i) |
| have hsP : (∑i∈s,P i)≪ξ := Measure.AbsolutelyContinuous.rfl.add_right _ |
| have hsQ : (∑i∈s,Q i)≪ξ := Measure.AbsolutelyContinuous.rfl.add_right' _ |
| have hp : ∀i∈s,P i≪ξ := fun i hi ↦ |
| (Measure.absolutelyContinuous_of_le (Finset.single_le_sum (fun _ _ ↦ bot_le) hi)).trans hsP |
| have hq : ∀i∈s,Q i≪ξ := fun i hi ↦ |
| (Measure.absolutelyContinuous_of_le (Finset.single_le_sum (fun _ _ ↦ bot_le) hi)).trans hsQ |
| calc |
| _ = ∑i∈s,affinity ξ (fun x ↦ ((P i).rnDeriv ξ x).toReal) |
| (fun x ↦ ((Q i).rnDeriv ξ x).toReal) := Finset.sum_congr rfl |
| (fun i hi ↦ measureAffinity_eq_dominated _ _ _ (hp i hi) (hq i hi)) |
| _ ≤ affinity ξ (fun x ↦ ∑i∈s,((P i).rnDeriv ξ x).toReal) |
| (fun x ↦ ∑i∈s,((Q i).rnDeriv ξ x).toReal) := |
| affinity_finset_sum s ξ _ _ (fun _ _ ↦ Measure.integrable_toReal_rnDeriv) |
| (fun _ _ ↦ Measure.integrable_toReal_rnDeriv) |
| (fun _ _ ↦ ENNReal.toReal_nonneg) (fun _ _ ↦ ENNReal.toReal_nonneg) |
| _ = _ := by |
| rw [measureAffinity_eq_dominated _ _ ξ hsP hsQ] |
| unfold affinity |
| apply integral_congr_ae |
| filter_upwards [rnDeriv_real_finset_sum s P ξ,rnDeriv_real_finset_sum s Q ξ] with x hx hy |
| rw [hx,hy] |
|
|
| lemma measureAffinity_smul (P Q : Measure α) [IsFiniteMeasure P] [IsFiniteMeasure Q] |
| (c : ℝ) (hc : 0≤c) : |
| measureAffinity (ENNReal.ofReal c • P) (ENNReal.ofReal c • Q)=c*measureAffinity P Q := by |
| let := P.smul_finite (c:=ENNReal.ofReal c) ENNReal.ofReal_ne_top |
| let := Q.smul_finite (c:=ENNReal.ofReal c) ENNReal.ofReal_ne_top |
| have hp : P≪P+Q := Measure.AbsolutelyContinuous.rfl.add_right Q |
| have hq : Q≪P+Q := Measure.AbsolutelyContinuous.rfl.add_right' P |
| rw [measureAffinity_eq_dominated _ _ (P+Q) (hp.smul_left _) (hq.smul_left _)] |
| unfold measureAffinity affinity |
| rw [← integral_const_mul] |
| apply integral_congr_ae |
| filter_upwards [Measure.rnDeriv_smul_left_of_ne_top P (P+Q) ENNReal.ofReal_ne_top, |
| Measure.rnDeriv_smul_left_of_ne_top Q (P+Q) ENNReal.ofReal_ne_top] with x hx hy |
| rw [hx,hy,Pi.smul_apply,Pi.smul_apply,smul_eq_mul,smul_eq_mul, |
| ENNReal.toReal_mul,ENNReal.toReal_mul,ENNReal.toReal_ofReal hc, |
| Real.sqrt_mul hc,Real.sqrt_mul hc] |
| have hs := Real.sq_sqrt hc |
| calc |
| _ = (Real.sqrt c)^2*(Real.sqrt ((P.rnDeriv (P+Q) x).toReal)* |
| Real.sqrt ((Q.rnDeriv (P+Q) x).toReal)) := by ring |
| _ = _ := by rw [hs] |
|
|
| lemma affinity_tag_normalization {ι : Type*} (s : Finset ι) |
| (P Q : Measure α) (Pg Qg : ι→Measure α) |
| [IsFiniteMeasure P] [IsFiniteMeasure Q] [∀i,IsFiniteMeasure (Pg i)] |
| [∀i,IsFiniteMeasure (Qg i)] (D : ℝ) (hD : 0<D) |
| (hp : ∑i∈s,Pg i ≤ ENNReal.ofReal D • P) |
| (hq : ∑i∈s,Qg i ≤ ENNReal.ofReal D • Q) : |
| D⁻¹*(∑i∈s,measureAffinity (Pg i) (Qg i)) ≤ measureAffinity P Q := by |
| let := P.smul_finite (c:=ENNReal.ofReal D) ENNReal.ofReal_ne_top |
| let := Q.smul_finite (c:=ENNReal.ofReal D) ENNReal.ofReal_ne_top |
| have h := (measureAffinity_finset_sum s Pg Qg).trans (measureAffinity_mono _ _ _ _ hp hq) |
| rw [measureAffinity_smul P Q D hD.le] at h |
| exact (inv_mul_le_iff₀ hD).2 h |
|
|
| end HardSphere.Comparison |
|
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| end |
|
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| end OAI |
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