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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 | 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
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