| import Mathlib |
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| namespace OAI |
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| section |
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| section |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal Topology BigOperators |
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| namespace HardSphere.Comparison |
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| variable {α : Type*} [MeasurableSpace α] {μ : Measure α} |
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| lemma integral_mul_le_sqrt (f g : α → ℝ) (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) |
| (hf0 : 0 ≤ᵐ[μ] f) (hg0 : 0 ≤ᵐ[μ] g) : |
| ∫ x, f x * g x ∂μ ≤ Real.sqrt (∫ x, f x ^ 2 ∂μ) * |
| Real.sqrt (∫ x, g x ^ 2 ∂μ) := by |
| have h := integral_mul_le_Lp_mul_Lq_of_nonneg |
| Real.HolderConjugate.two_two hf0 hg0 (by simpa using hf) (by simpa using hg) |
| simpa only [Real.rpow_two, ← Real.sqrt_eq_rpow] using h |
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| lemma memLp_sqrt {f : α → ℝ} (hf : Integrable f μ) (hf0 : 0 ≤ᵐ[μ] f) : |
| MemLp (fun x ↦ Real.sqrt (f x)) 2 μ := by |
| refine (memLp_two_iff_integrable_sq (Real.continuous_sqrt.comp_aestronglyMeasurable hf.1)).2 ?_ |
| apply hf.congr |
| filter_upwards [hf0] with x hx |
| exact (Real.sq_sqrt hx).symm |
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| def affinity (μ : Measure α) (p q : α → ℝ) : ℝ := |
| ∫ x, Real.sqrt (p x) * Real.sqrt (q x) ∂μ |
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| lemma affinity_nonneg (p q : α → ℝ) : 0 ≤ affinity μ p q := by |
| exact integral_nonneg fun x ↦ mul_nonneg (Real.sqrt_nonneg _) (Real.sqrt_nonneg _) |
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| lemma affinity_integrable {p q : α → ℝ} (hp : Integrable p μ) (hq : Integrable q μ) |
| (hp0 : 0 ≤ᵐ[μ] p) (hq0 : 0 ≤ᵐ[μ] q) : |
| Integrable (fun x ↦ Real.sqrt (p x) * Real.sqrt (q x)) μ := by |
| exact (memLp_sqrt hp hp0).integrable_mul (memLp_sqrt hq hq0) |
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| lemma sqrt_ratio_product {p q s : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) (hs : 0 ≤ s) |
| (hsp : p = 0 → s = 0) (hsq : q = 0 → s = 0) : |
| Real.sqrt (s / Real.sqrt p * (s / Real.sqrt q)) * |
| Real.sqrt (Real.sqrt p * Real.sqrt q) = s := by |
| by_cases hpz : p = 0 |
| · simp [hpz, hsp hpz] |
| by_cases hqz : q = 0 |
| · simp [hqz, hsq hqz] |
| have hp' : Real.sqrt p ≠ 0 := ne_of_gt (Real.sqrt_pos.2 (lt_of_le_of_ne hp (Ne.symm hpz))) |
| have hq' : Real.sqrt q ≠ 0 := ne_of_gt (Real.sqrt_pos.2 (lt_of_le_of_ne hq (Ne.symm hqz))) |
| have hprod : (s / Real.sqrt p * (s / Real.sqrt q)) * |
| (Real.sqrt p * Real.sqrt q) = s ^ 2 := by |
| field_simp |
| rw [← Real.sqrt_mul (mul_nonneg (div_nonneg hs (Real.sqrt_nonneg p)) |
| (div_nonneg hs (Real.sqrt_nonneg q))), hprod, Real.sqrt_sq hs] |
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| lemma common_density_bound {p q s : α → ℝ} |
| (hp : Integrable p μ) (hq : Integrable q μ) |
| (hp0 : 0 ≤ᵐ[μ] p) (hq0 : 0 ≤ᵐ[μ] q) (hs0 : 0 ≤ᵐ[μ] s) |
| (hsp : ∀ᵐ x ∂μ, p x = 0 → s x = 0) |
| (hsq : ∀ᵐ x ∂μ, q x = 0 → s x = 0) |
| (hP : MemLp (fun x ↦ s x / Real.sqrt (p x)) 2 μ) |
| (hQ : MemLp (fun x ↦ s x / Real.sqrt (q x)) 2 μ) : |
| (∫ x, s x ∂μ) ^ 2 ≤ |
| Real.sqrt (∫ x, (s x / Real.sqrt (p x)) ^ 2 ∂μ) * |
| Real.sqrt (∫ x, (s x / Real.sqrt (q x)) ^ 2 ∂μ) * affinity μ p q := by |
| let a := fun x ↦ s x / Real.sqrt (p x) |
| let b := fun x ↦ s x / Real.sqrt (q x) |
| let c := fun x ↦ Real.sqrt (p x) * Real.sqrt (q x) |
| have ha0 : 0 ≤ᵐ[μ] a := hs0.mono fun x hx ↦ div_nonneg hx (Real.sqrt_nonneg _) |
| have hb0 : 0 ≤ᵐ[μ] b := hs0.mono fun x hx ↦ div_nonneg hx (Real.sqrt_nonneg _) |
| have hab : Integrable (fun x ↦ a x * b x) μ := hP.integrable_mul hQ |
| have hab0 : 0 ≤ᵐ[μ] (fun x ↦ a x * b x) := by |
| filter_upwards [ha0, hb0] with x hx hy using mul_nonneg hx hy |
| have hc : Integrable c μ := affinity_integrable hp hq hp0 hq0 |
| have hc0 : 0 ≤ᵐ[μ] c := Eventually.of_forall fun x ↦ by dsimp [c]; positivity |
| have hs : (fun x ↦ Real.sqrt (a x * b x) * Real.sqrt (c x)) =ᵐ[μ] s := by |
| filter_upwards [hp0, hq0, hs0, hsp, hsq] with x hp' hq' hs' hsp' hsq' |
| exact sqrt_ratio_product hp' hq' hs' hsp' hsq' |
| have h₁ := integral_mul_le_sqrt (fun x ↦ Real.sqrt (a x * b x)) |
| (fun x ↦ Real.sqrt (c x)) (memLp_sqrt hab hab0) (memLp_sqrt hc hc0) |
| (Eventually.of_forall fun x ↦ Real.sqrt_nonneg _) (Eventually.of_forall fun x ↦ Real.sqrt_nonneg _) |
| have habsq : (∫ x, Real.sqrt (a x * b x) ^ 2 ∂μ) = ∫ x, a x * b x ∂μ := by |
| apply integral_congr_ae |
| exact hab0.mono fun _ h ↦ Real.sq_sqrt h |
| have hcsq : (∫ x, Real.sqrt (c x) ^ 2 ∂μ) = affinity μ p q := by |
| apply integral_congr_ae |
| exact hc0.mono fun _ h ↦ Real.sq_sqrt h |
| rw [integral_congr_ae hs, habsq, hcsq] at h₁ |
| have h₂ : (∫ x, s x ∂μ) ^ 2 ≤ (∫ x, a x * b x ∂μ) * affinity μ p q := by |
| have h := (sq_le_sq₀ (integral_nonneg_of_ae hs0) (by positivity)).2 h₁ |
| simpa only [mul_pow, Real.sq_sqrt (integral_nonneg_of_ae hab0), |
| Real.sq_sqrt (affinity_nonneg p q)] using h |
| exact h₂.trans (mul_le_mul_of_nonneg_right |
| (integral_mul_le_sqrt a b hP hQ ha0 hb0) (affinity_nonneg p q)) |
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| lemma ratio_sqrt_sq {s p : ℝ} (hp : 0 ≤ p) : |
| (s / Real.sqrt p) ^ 2 = s ^ 2 / p := by |
| rw [div_pow, Real.sq_sqrt hp] |
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| lemma memLp_ratio_sqrt {s p : α → ℝ} (hs : AEStronglyMeasurable s μ) |
| (hp : AEStronglyMeasurable p μ) (hp0 : 0 ≤ᵐ[μ] p) |
| (h : Integrable (fun x ↦ s x ^ 2 / p x) μ) : |
| MemLp (fun x ↦ s x / Real.sqrt (p x)) 2 μ := by |
| refine (memLp_two_iff_integrable_sq |
| ((hs.aemeasurable.div (Real.continuous_sqrt.measurable.comp_aemeasurable hp.aemeasurable)).aestronglyMeasurable)).2 ?_ |
| apply h.congr |
| exact hp0.mono fun x hx ↦ (ratio_sqrt_sq hx).symm |
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| theorem common_density_affinity {p q s : α → ℝ} {C_P C_Q : ℝ} |
| (hp : Integrable p μ) (hq : Integrable q μ) (hs : Integrable s μ) |
| (hp0 : 0 ≤ᵐ[μ] p) (hq0 : 0 ≤ᵐ[μ] q) (hs0 : 0 ≤ᵐ[μ] s) |
| (hsp : ∀ᵐ x ∂μ, p x = 0 → s x = 0) |
| (hsq : ∀ᵐ x ∂μ, q x = 0 → s x = 0) |
| (hPi : Integrable (fun x ↦ s x ^ 2 / p x) μ) |
| (hQi : Integrable (fun x ↦ s x ^ 2 / q x) μ) |
| (hP : (∫ x, s x ^ 2 / p x ∂μ) ≤ C_P) |
| (hQ : (∫ x, s x ^ 2 / q x ∂μ) ≤ C_Q) |
| (hm : 0 < ∫ x, s x ∂μ) : |
| (∫ x, s x ∂μ) ^ 2 / Real.sqrt (C_P * C_Q) ≤ affinity μ p q := by |
| have hP' := memLp_ratio_sqrt hs.1 hp.1 hp0 hPi |
| have hQ' := memLp_ratio_sqrt hs.1 hq.1 hq0 hQi |
| have hPeq : (∫ x, (s x / Real.sqrt (p x)) ^ 2 ∂μ) = |
| ∫ x, s x ^ 2 / p x ∂μ := |
| integral_congr_ae (hp0.mono fun x hx ↦ ratio_sqrt_sq hx) |
| have hQeq : (∫ x, (s x / Real.sqrt (q x)) ^ 2 ∂μ) = |
| ∫ x, s x ^ 2 / q x ∂μ := |
| integral_congr_ae (hq0.mono fun x hx ↦ ratio_sqrt_sq hx) |
| have hP0 : 0 ≤ ∫ x, s x ^ 2 / p x ∂μ := by |
| rw [← hPeq] |
| exact integral_nonneg fun x ↦ sq_nonneg _ |
| have hQ0 : 0 ≤ ∫ x, s x ^ 2 / q x ∂μ := by |
| rw [← hQeq] |
| exact integral_nonneg fun x ↦ sq_nonneg _ |
| have hCP0 := hP0.trans hP |
| have hCQ0 := hQ0.trans hQ |
| have h := common_density_bound hp hq hp0 hq0 hs0 hsp hsq hP' hQ' |
| rw [hPeq, hQeq] at h |
| have h' : (∫ x, s x ∂μ) ^ 2 ≤ Real.sqrt (C_P * C_Q) * affinity μ p q := by |
| rw [Real.sqrt_mul hCP0] |
| exact h.trans (mul_le_mul_of_nonneg_right |
| (mul_le_mul (Real.sqrt_le_sqrt hP) (Real.sqrt_le_sqrt hQ) |
| (Real.sqrt_nonneg _) (Real.sqrt_nonneg _)) (affinity_nonneg p q)) |
| have hden : 0 < Real.sqrt (C_P * C_Q) := by |
| by_contra hn |
| have hz : Real.sqrt (C_P * C_Q) = 0 := |
| le_antisymm (le_of_not_gt hn) (Real.sqrt_nonneg _) |
| rw [hz, zero_mul] at h' |
| nlinarith |
| exact (div_le_iff₀ hden).2 (by simpa only [mul_comm] using h') |
| |
| def measureAffinity (P Q : Measure α) : ℝ := |
| affinity (P + Q) (fun x ↦ (P.rnDeriv (P + Q) x).toReal) |
| (fun x ↦ (Q.rnDeriv (P + Q) x).toReal) |
| |
| lemma real_rnDeriv_chain {ν P ξ : Measure α} |
| [IsFiniteMeasure ν] [IsFiniteMeasure P] [IsFiniteMeasure ξ] (hνP : ν ≪ P) : |
| (fun x ↦ (ν.rnDeriv P x).toReal * (P.rnDeriv ξ x).toReal) =ᵐ[ξ] |
| (fun x ↦ (ν.rnDeriv ξ x).toReal) := by |
| filter_upwards [Measure.rnDeriv_mul_rnDeriv (κ := ξ) hνP] with x hx |
| exact ENNReal.toReal_mul.symm.trans (congrArg ENNReal.toReal hx) |
| |
| lemma sq_mul_div (a b : ℝ) : (a * b) ^ 2 / b = b * a ^ 2 := by |
| by_cases hb : b = 0 |
| · simp [hb] |
| · field_simp |
| |
| lemma rnDeriv_density_moment {ν P ξ : Measure α} |
| [IsFiniteMeasure ν] [IsFiniteMeasure P] [IsFiniteMeasure ξ] |
| (hνP : ν ≪ P) (hPξ : P ≪ ξ) : |
| (∫ x, (ν.rnDeriv ξ x).toReal ^ 2 / (P.rnDeriv ξ x).toReal ∂ξ) = |
| ∫ x, (ν.rnDeriv P x).toReal ^ 2 ∂P := by |
| calc |
| (∫ x, (ν.rnDeriv ξ x).toReal ^ 2 / (P.rnDeriv ξ x).toReal ∂ξ) = |
| ∫ x, (P.rnDeriv ξ x).toReal * (ν.rnDeriv P x).toReal ^ 2 ∂ξ := by |
| apply integral_congr_ae |
| filter_upwards [real_rnDeriv_chain (ξ := ξ) hνP] with x hx |
| rw [← hx, sq_mul_div] |
| _ = _ := integral_toReal_rnDeriv_mul hPξ |
| |
| lemma integrable_rnDeriv_density_moment {ν P ξ : Measure α} |
| [IsFiniteMeasure ν] [IsFiniteMeasure P] [IsFiniteMeasure ξ] |
| (hνP : ν ≪ P) (hPξ : P ≪ ξ) |
| (h : Integrable (fun x ↦ (ν.rnDeriv P x).toReal ^ 2) P) : |
| Integrable (fun x ↦ (ν.rnDeriv ξ x).toReal ^ 2 / (P.rnDeriv ξ x).toReal) ξ := by |
| apply ((integrable_toReal_rnDeriv_mul_iff hPξ).2 h).congr |
| filter_upwards [real_rnDeriv_chain (ξ := ξ) hνP] with x hx |
| rw [← hx, sq_mul_div] |
| |
| theorem common_measure_affinity {ν P Q : Measure α} |
| [IsFiniteMeasure ν] [IsFiniteMeasure P] [IsFiniteMeasure Q] |
| {C_P C_Q : ℝ} (hνP : ν ≪ P) (hνQ : ν ≪ Q) |
| (hm : 0 < ν.real univ) |
| (hPi : Integrable (fun x ↦ (ν.rnDeriv P x).toReal ^ 2) P) |
| (hQi : Integrable (fun x ↦ (ν.rnDeriv Q x).toReal ^ 2) Q) |
| (hP : (∫ x, (ν.rnDeriv P x).toReal ^ 2 ∂P) ≤ C_P) |
| (hQ : (∫ x, (ν.rnDeriv Q x).toReal ^ 2 ∂Q) ≤ C_Q) : |
| (ν.real univ) ^ 2 / Real.sqrt (C_P * C_Q) ≤ measureAffinity P Q := by |
| have hPξ : P ≪ P + Q := Measure.AbsolutelyContinuous.rfl.add_right Q |
| have hQξ : Q ≪ P + Q := Measure.AbsolutelyContinuous.rfl.add_right' P |
| have hνξ := hνP.trans hPξ |
| have chainP := real_rnDeriv_chain (ξ := P + Q) hνP |
| have chainQ := real_rnDeriv_chain (ξ := P + Q) hνQ |
| have hsp : ∀ᵐ x ∂(P + Q), (P.rnDeriv (P + Q) x).toReal = 0 → |
| (ν.rnDeriv (P + Q) x).toReal = 0 := by |
| filter_upwards [chainP] with x hx hz |
| rw [← hx, hz, mul_zero] |
| have hsq : ∀ᵐ x ∂(P + Q), (Q.rnDeriv (P + Q) x).toReal = 0 → |
| (ν.rnDeriv (P + Q) x).toReal = 0 := by |
| filter_upwards [chainQ] with x hx hz |
| rw [← hx, hz, mul_zero] |
| have h := common_density_affinity (C_P := C_P) (C_Q := C_Q) |
| (μ := P + Q) (p := fun x ↦ (P.rnDeriv (P + Q) x).toReal) |
| (q := fun x ↦ (Q.rnDeriv (P + Q) x).toReal) |
| (s := fun x ↦ (ν.rnDeriv (P + Q) x).toReal) |
| Measure.integrable_toReal_rnDeriv Measure.integrable_toReal_rnDeriv |
| Measure.integrable_toReal_rnDeriv |
| (Eventually.of_forall fun _ ↦ ENNReal.toReal_nonneg) |
| (Eventually.of_forall fun _ ↦ ENNReal.toReal_nonneg) |
| (Eventually.of_forall fun _ ↦ ENNReal.toReal_nonneg) |
| hsp hsq |
| (integrable_rnDeriv_density_moment hνP hPξ hPi) |
| (integrable_rnDeriv_density_moment hνQ hQξ hQi) |
| (by rwa [rnDeriv_density_moment hνP hPξ]) |
| (by rwa [rnDeriv_density_moment hνQ hQξ]) |
| (by rwa [Measure.integral_toReal_rnDeriv hνξ]) |
| simpa only [Measure.integral_toReal_rnDeriv hνξ, measureAffinity] using h |
|
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| end HardSphere.Comparison |
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| end |
| end |
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| end |
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| end OAI |
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