| import OAI.Probability.FreeIsing.Trees.CoupledPathProducts | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| section Probability | |
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} | |
| theorem integral_abs_mul_le_sqrt_moments {U Z : Ω → ℝ} | |
| (hU : MemLp U 2 μ) (hZ : MemLp Z 2 μ) : | |
| (∫ ω, |U ω| * |Z ω| ∂μ) ≤ | |
| Real.sqrt (∫ ω, U ω ^ 2 ∂μ) * Real.sqrt (∫ ω, Z ω ^ 2 ∂μ) := by | |
| have hpq : (2 : ℝ).HolderConjugate 2 := by | |
| norm_num [Real.holderConjugate_iff] | |
| have h := integral_mul_norm_le_Lp_mul_Lq hpq (by simpa using hU) (by simpa using hZ) | |
| simpa only [Real.norm_eq_abs, Real.rpow_two, sq_abs, Real.sqrt_eq_rpow] using h | |
| theorem integral_weighted_abs_mul_le {U Z w : Ω → ℝ} {C : ℝ} | |
| (hU : MemLp U 2 μ) (hZ : MemLp Z 2 μ) (hw : Measurable w) | |
| (hw0 : ∀ ω, 0 ≤ w ω) (hwC : ∀ ω, w ω ≤ C) : | |
| Integrable (fun ω => |U ω| * w ω * |Z ω|) μ ∧ | |
| (∫ ω, |U ω| * w ω * |Z ω| ∂μ) ≤ | |
| Real.sqrt (C * ∫ ω, U ω ^ 2 ∂μ) * | |
| Real.sqrt (∫ ω, w ω * Z ω ^ 2 ∂μ) := by | |
| let F := fun ω => U ω * Real.sqrt (w ω) | |
| let G := fun ω => Z ω * Real.sqrt (w ω) | |
| have hwt : MemLp (fun ω => Real.sqrt (w ω)) (⊤ : ENNReal) μ := by | |
| apply memLp_top_of_bound hw.sqrt.aestronglyMeasurable (Real.sqrt C) | |
| filter_upwards [] with ω | |
| rw [Real.norm_eq_abs, abs_of_nonneg (Real.sqrt_nonneg _)] | |
| exact Real.sqrt_le_sqrt (hwC ω) | |
| have hF : MemLp F 2 μ := hU.mul hwt | |
| have hG : MemLp G 2 μ := hZ.mul hwt | |
| have hFsq (ω : Ω) : F ω ^ 2 = w ω * U ω ^ 2 := by | |
| dsimp [F] | |
| rw [mul_pow, Real.sq_sqrt (hw0 ω), mul_comm] | |
| have hGsq (ω : Ω) : G ω ^ 2 = w ω * Z ω ^ 2 := by | |
| dsimp [G] | |
| rw [mul_pow, Real.sq_sqrt (hw0 ω), mul_comm] | |
| have hprod (ω : Ω) : |F ω| * |G ω| = |U ω| * w ω * |Z ω| := by | |
| dsimp [F, G] | |
| rw [abs_mul, abs_mul, abs_of_nonneg (Real.sqrt_nonneg _)] | |
| calc | |
| |U ω| * Real.sqrt (w ω) * (|Z ω| * Real.sqrt (w ω)) = | |
| |U ω| * |Z ω| * (Real.sqrt (w ω)) ^ 2 := by ring | |
| _ = |U ω| * w ω * |Z ω| := by rw [Real.sq_sqrt (hw0 ω)]; ring | |
| have hI : Integrable (fun ω => |U ω| * w ω * |Z ω|) μ := by | |
| have h := hF.norm.integrable_mul hG.norm | |
| change Integrable (fun ω => ‖F ω‖ * ‖G ω‖) μ at h | |
| simpa only [Real.norm_eq_abs, Pi.mul_apply, hprod] using h | |
| have hcs := integral_abs_mul_le_sqrt_moments hF hG | |
| simp_rw [hprod, hFsq, hGsq] at hcs | |
| have hFbound : (∫ ω, w ω * U ω ^ 2 ∂μ) ≤ C * ∫ ω, U ω ^ 2 ∂μ := by | |
| have hint : Integrable (fun ω => w ω * U ω ^ 2) μ := by | |
| simpa only [hFsq] using hF.integrable_sq | |
| calc | |
| (∫ ω, w ω * U ω ^ 2 ∂μ) ≤ ∫ ω, C * U ω ^ 2 ∂μ := | |
| integral_mono hint (hU.integrable_sq.const_mul C) | |
| (fun ω => mul_le_mul_of_nonneg_right (hwC ω) (sq_nonneg _)) | |
| _ = C * ∫ ω, U ω ^ 2 ∂μ := integral_const_mul _ _ | |
| exact ⟨hI, hcs.trans (mul_le_mul_of_nonneg_right | |
| (Real.sqrt_le_sqrt hFbound) (Real.sqrt_nonneg _))⟩ | |
| theorem integral_abs_mul_prod_responseAttenuation_sq_sub_sq_le | |
| {ι : Type*} (s : Finset ι) {theta : ℝ} (hθ : theta ^ 2 ≤ 1) | |
| {X Y : ι → Ω → ℝ} {U : Ω → ℝ} | |
| (hX : ∀ i ∈ s, Measurable (X i)) (hY : ∀ i ∈ s, Measurable (Y i)) | |
| (hInt : ∀ i ∈ s, Integrable (fun ω => (X i ω - Y i ω) ^ 2) μ) | |
| (hU : MemLp U 2 μ) : | |
| let P := fun ω => ∏ i ∈ s, responseAttenuation theta (X i ω) | |
| let P' := fun ω => ∏ i ∈ s, responseAttenuation theta (Y i ω) | |
| let Z := fun ω => ∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) - | |
| Real.log (responseAttenuation theta (Y i ω))) | |
| Integrable (fun ω => |U ω| * |P ω ^ 2 - P' ω ^ 2|) μ ∧ | |
| (∫ ω, |U ω| * |P ω ^ 2 - P' ω ^ 2| ∂μ) ≤ | |
| 2 * Real.sqrt (2 * ∫ ω, U ω ^ 2 ∂μ) * | |
| Real.sqrt (∫ ω, (P ω ^ 2 + P' ω ^ 2) * Z ω ^ 2 ∂μ) := by | |
| dsimp only | |
| let P := fun ω => ∏ i ∈ s, responseAttenuation theta (X i ω) | |
| let P' := fun ω => ∏ i ∈ s, responseAttenuation theta (Y i ω) | |
| let Z := fun ω => ∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) - | |
| Real.log (responseAttenuation theta (Y i ω))) | |
| let w := fun ω => P ω ^ 2 + P' ω ^ 2 | |
| have hP : Measurable P := measurable_prod_responseAttenuation s theta hX | |
| have hP' : Measurable P' := measurable_prod_responseAttenuation s theta hY | |
| have hw : Measurable w := (hP.pow_const 2).add (hP'.pow_const 2) | |
| have hw0 (ω : Ω) : 0 ≤ w ω := add_nonneg (sq_nonneg _) (sq_nonneg _) | |
| have hw2 (ω : Ω) : w ω ≤ 2 := by | |
| have hp := prod_responseAttenuation_pos s (fun i => X i ω) hθ | |
| have hp' := prod_responseAttenuation_pos s (fun i => Y i ω) hθ | |
| have hl := prod_attenuation_le_one s (fun i => X i ω) hθ | |
| have hl' := prod_attenuation_le_one s (fun i => Y i ω) hθ | |
| change P ω > 0 at hp | |
| change P' ω > 0 at hp' | |
| change P ω ≤ 1 at hl | |
| change P' ω ≤ 1 at hl' | |
| dsimp only [w] | |
| nlinarith | |
| have hZ : MemLp Z 2 μ := memLp_sum_log_responseAttenuation_sub s hθ hX hY hInt | |
| have hbound := integral_weighted_abs_mul_le hU hZ hw hw0 hw2 | |
| have hpoint (ω : Ω) : |U ω| * |P ω ^ 2 - P' ω ^ 2| ≤ | |
| 2 * (|U ω| * w ω * |Z ω|) := by | |
| calc | |
| |U ω| * |P ω ^ 2 - P' ω ^ 2| ≤ |U ω| * (2 * w ω * |Z ω|) := | |
| mul_le_mul_of_nonneg_left | |
| (abs_prod_responseAttenuation_sq_sub_sq_le s | |
| (fun i => X i ω) (fun i => Y i ω) hθ) (abs_nonneg _) | |
| _ = _ := by ring | |
| have hDiff : Integrable (fun ω => |U ω| * |P ω ^ 2 - P' ω ^ 2|) μ := by | |
| apply (hbound.1.const_mul 2).mono' | |
| (hU.aestronglyMeasurable.aemeasurable.abs.aestronglyMeasurable.mul | |
| (((hP.pow_const 2).sub (hP'.pow_const 2)).abs.aestronglyMeasurable)) | |
| filter_upwards [] with ω | |
| rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (abs_nonneg _) (abs_nonneg _))] | |
| exact hpoint ω | |
| refine ⟨hDiff, ?_⟩ | |
| calc | |
| (∫ ω, |U ω| * |P ω ^ 2 - P' ω ^ 2| ∂μ) ≤ | |
| ∫ ω, 2 * (|U ω| * w ω * |Z ω|) ∂μ := | |
| integral_mono hDiff (hbound.1.const_mul 2) hpoint | |
| _ = 2 * ∫ ω, |U ω| * w ω * |Z ω| ∂μ := integral_const_mul _ _ | |
| _ ≤ 2 * (Real.sqrt (2 * ∫ ω, U ω ^ 2 ∂μ) * | |
| Real.sqrt (∫ ω, w ω * Z ω ^ 2 ∂μ)) := | |
| mul_le_mul_of_nonneg_left hbound.2 (by norm_num) | |
| _ = _ := by ring | |
| theorem integral_mul_prod_responseAttenuation_sq_le | |
| {ι : Type*} (s : Finset ι) {theta : ℝ} (hθ : theta ^ 2 ≤ 1) | |
| {X : ι → Ω → ℝ} {U : Ω → ℝ} | |
| (hX : ∀ i ∈ s, Measurable (X i)) (hU : MemLp U 2 μ) : | |
| Integrable (fun ω => U ω ^ 2 * | |
| (∏ i ∈ s, responseAttenuation theta (X i ω)) ^ 2) μ ∧ | |
| (∫ ω, U ω ^ 2 * (∏ i ∈ s, responseAttenuation theta (X i ω)) ^ 2 ∂μ) ≤ | |
| ∫ ω, U ω ^ 2 ∂μ := by | |
| let P := fun ω => ∏ i ∈ s, responseAttenuation theta (X i ω) | |
| have hP : Measurable P := measurable_prod_responseAttenuation s theta hX | |
| have hpoint (ω : Ω) : U ω ^ 2 * P ω ^ 2 ≤ U ω ^ 2 := by | |
| have hp := prod_responseAttenuation_pos s (fun i => X i ω) hθ | |
| have hl := prod_attenuation_le_one s (fun i => X i ω) hθ | |
| change 0 < P ω at hp | |
| change P ω ≤ 1 at hl | |
| have hsq : P ω ^ 2 ≤ 1 := by nlinarith | |
| simpa using mul_le_mul_of_nonneg_left hsq (sq_nonneg (U ω)) | |
| have hint : Integrable (fun ω => U ω ^ 2 * P ω ^ 2) μ := by | |
| apply hU.integrable_sq.mono' | |
| ((hU.aestronglyMeasurable.pow 2).mul ((hP.pow_const 2).aestronglyMeasurable)) | |
| filter_upwards [] with ω | |
| rw [Real.norm_eq_abs, abs_of_nonneg (mul_nonneg (sq_nonneg _) (sq_nonneg _))] | |
| exact hpoint ω | |
| exact ⟨hint, integral_mono hint hU.integrable_sq hpoint⟩ | |
| end Probability | |
| end Problem367 | |
| end | |
| end OAI | |