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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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory Set Filter
open scoped Topology BigOperators ENNReal
namespace Problem367.CompensatorVariation
def squaredIncrements (a : ℕ → ℝ) (N : ℕ) : ℝ :=
∑ i ∈ Finset.range N, (a (i + 1) - a i) ^ 2
theorem squaredIncrements_nonneg (a : ℕ → ℝ) (N : ℕ) :
0 ≤ squaredIncrements a N := Finset.sum_nonneg (fun _ _ => sq_nonneg _)
theorem squaredIncrements_le {a : ℕ → ℝ} {N : ℕ} {δ : ℝ}
(hpos : ∀ i < N, 0 ≤ a (i + 1) - a i)
(hsmall : ∀ i < N, a (i + 1) - a i ≤ δ) :
squaredIncrements a N ≤ δ * (a N - a 0) := by
calc
squaredIncrements a N ≤ ∑ i ∈ Finset.range N, δ * (a (i + 1) - a i) := by
apply Finset.sum_le_sum
intro i hi
rw [pow_two]
exact mul_le_mul_of_nonneg_right (hsmall i (Finset.mem_range.mp hi))
(hpos i (Finset.mem_range.mp hi))
_ = δ * (a N - a 0) := by rw [← Finset.mul_sum, Finset.sum_range_sub]
theorem squaredIncrements_le_total_sq {a : ℕ → ℝ} {N : ℕ}
(hpos : ∀ i < N, 0 ≤ a (i + 1) - a i) :
squaredIncrements a N ≤ (a N - a 0) ^ 2 := by
have h := Finset.sum_sq_le_sq_sum_of_nonneg
(s := Finset.range N) (fun i hi => hpos i (Finset.mem_range.mp hi))
simpa only [squaredIncrements, Finset.sum_range_sub] using h
theorem total_nonneg {a : ℕ → ℝ} {N : ℕ}
(hpos : ∀ i < N, 0 ≤ a (i + 1) - a i) : 0 ≤ a N - a 0 := by
rw [← Finset.sum_range_sub a N]
exact Finset.sum_nonneg (fun i hi => hpos i (Finset.mem_range.mp hi))
theorem increment_le_total {a : ℕ → ℝ} {N : ℕ}
(hpos : ∀ i < N, 0 ≤ a (i + 1) - a i) {i : ℕ} (hi : i < N) :
a (i + 1) - a i ≤ a N - a 0 := by
rw [← Finset.sum_range_sub a N]
exact Finset.single_le_sum (fun j hj => hpos j (Finset.mem_range.mp hj))
(Finset.mem_range.mpr hi)
theorem tendsto_squaredIncrements {I : Type*} [PseudoMetricSpace I]
{K : Set I} (hK : IsCompact K) {A : I → ℝ} (hA : ContinuousOn A K)
(t : ℕ → ℕ → I) (N : ℕ → ℕ) (mesh : ℕ → ℝ)
(hmem : ∀ n i, i ≤ N n → t n i ∈ K)
(hdist : ∀ n i, i < N n → dist (t n i) (t n (i + 1)) ≤ mesh n)
(hmesh : Tendsto mesh atTop (𝓝 0))
(hpos : ∀ n i, i < N n → 0 ≤ A (t n (i + 1)) - A (t n i))
{C : ℝ} (hC : 0 ≤ C) (hbound : ∀ n, A (t n (N n)) - A (t n 0) ≤ C) :
Tendsto (fun n => squaredIncrements (fun i => A (t n i)) (N n)) atTop (𝓝 0) := by
apply Metric.tendsto_atTop.mpr
intro ε hε
let η := ε / (C + 1)
have hη : 0 < η := div_pos hε (by linarith)
obtain ⟨δ, hδ, hmod⟩ := Metric.uniformContinuousOn_iff.mp
(hK.uniformContinuousOn_of_continuous hA) η hη
obtain ⟨n₀, hn₀⟩ := (hmesh.eventually (gt_mem_nhds hδ)).exists_forall_of_atTop
refine ⟨n₀, ?_⟩
intro n hn
have hsmall (i : ℕ) (hi : i < N n) : A (t n (i + 1)) - A (t n i) ≤ η := by
have hh := hmod (t n (i + 1)) (hmem n (i + 1) (Nat.succ_le_of_lt hi))
(t n i) (hmem n i hi.le) (by
rw [dist_comm]
exact (hdist n i hi).trans_lt (hn₀ n hn))
exact (le_abs_self _).trans (by simpa only [Real.dist_eq] using hh.le)
rw [Real.dist_eq, sub_zero, abs_of_nonneg (squaredIncrements_nonneg _ _)]
calc
squaredIncrements (fun i => A (t n i)) (N n) ≤
η * (A (t n (N n)) - A (t n 0)) := squaredIncrements_le (hpos n) hsmall
_ ≤ η * C := mul_le_mul_of_nonneg_left (hbound n) hη.le
_ < ε := by
change ε / (C + 1) * C < ε
rw [div_mul_eq_mul_div]
apply (div_lt_iff₀ (by linarith : 0 < C + 1)).2
nlinarith
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
theorem memLp_increment_of_total_bound [IsFiniteMeasure μ]
{a : ℕ → Ω → ℝ} (hmeas : ∀ i, AEStronglyMeasurable (a i) μ) {N : ℕ}
(hpos : ∀ᵐ ω ∂μ, ∀ i < N, 0 ≤ a (i + 1) ω - a i ω)
{C : ℝ} (hbound : ∀ᵐ ω ∂μ, a N ω - a 0 ω ≤ C)
{i : ℕ} (hi : i < N) (p : ℝ≥0∞) :
MemLp (fun ω => a (i + 1) ω - a i ω) p μ := by
apply MemLp.of_bound ((hmeas _).sub (hmeas _)) C
filter_upwards [hpos, hbound] with ω hp hb
rw [Real.norm_eq_abs, Pi.sub_apply, abs_of_nonneg (hp i hi)]
exact (increment_le_total (a := fun j => a j ω) hp hi).trans hb
theorem tendsto_integral_squaredIncrements [IsFiniteMeasure μ]
{I : Type*} [PseudoMetricSpace I] {K : Set I} (hK : IsCompact K)
{A : I → Ω → ℝ} (hmeas : ∀ t, AEStronglyMeasurable (A t) μ)
(hc : ∀ᵐ ω ∂μ, ContinuousOn (fun t => A t ω) K)
(t : ℕ → ℕ → I) (N : ℕ → ℕ) (mesh : ℕ → ℝ)
(hmem : ∀ n i, i ≤ N n → t n i ∈ K)
(hdist : ∀ n i, i < N n → dist (t n i) (t n (i + 1)) ≤ mesh n)
(hmesh : Tendsto mesh atTop (𝓝 0))
(hpos : ∀ᵐ ω ∂μ, ∀ n i, i < N n → 0 ≤ A (t n (i + 1)) ω - A (t n i) ω)
{C : ℝ} (hC : 0 ≤ C)
(hbound : ∀ᵐ ω ∂μ, ∀ n, A (t n (N n)) ω - A (t n 0) ω ≤ C) :
Tendsto (fun n => ∫ ω, squaredIncrements (fun i => A (t n i) ω) (N n) ∂μ)
atTop (𝓝 0) := by
have hm (n : ℕ) : AEStronglyMeasurable
(fun ω => squaredIncrements (fun i => A (t n i) ω) (N n)) μ := by
convert (Finset.aestronglyMeasurable_sum (Finset.range (N n))
(fun i _ => ((hmeas (t n (i + 1))).sub (hmeas (t n i))).pow 2)) using 1
ext ω
simp only [squaredIncrements, Finset.sum_apply, Pi.pow_apply, Pi.sub_apply]
have hh := tendsto_integral_of_dominated_convergence
(μ := μ) (F := fun n ω => squaredIncrements (fun i => A (t n i) ω) (N n))
(f := fun _ => (0 : ℝ)) (fun _ => C ^ 2) hm (integrable_const _) ?_ ?_
· simpa only [integral_zero] using hh
· intro n
filter_upwards [hpos, hbound] with ω hp hb
rw [Real.norm_eq_abs, abs_of_nonneg (squaredIncrements_nonneg _ _)]
exact (squaredIncrements_le_total_sq (hp n)).trans
((sq_le_sq₀ (total_nonneg (a := fun i => A (t n i) ω) (hp n)) hC).2 (hb n))
· filter_upwards [hc, hpos, hbound] with ω hω hp hb
exact tendsto_squaredIncrements hK hω t N mesh hmem hdist hmesh hp hC hb
theorem tendsto_sum_integral_sq_increments [IsFiniteMeasure μ]
{I : Type*} [PseudoMetricSpace I] {K : Set I} (hK : IsCompact K)
{A : I → Ω → ℝ} (hmeas : ∀ t, AEStronglyMeasurable (A t) μ)
(hc : ∀ᵐ ω ∂μ, ContinuousOn (fun t => A t ω) K)
(t : ℕ → ℕ → I) (N : ℕ → ℕ) (mesh : ℕ → ℝ)
(hmem : ∀ n i, i ≤ N n → t n i ∈ K)
(hdist : ∀ n i, i < N n → dist (t n i) (t n (i + 1)) ≤ mesh n)
(hmesh : Tendsto mesh atTop (𝓝 0))
(hpos : ∀ᵐ ω ∂μ, ∀ n i, i < N n → 0 ≤ A (t n (i + 1)) ω - A (t n i) ω)
{C : ℝ} (hC : 0 ≤ C)
(hbound : ∀ᵐ ω ∂μ, ∀ n, A (t n (N n)) ω - A (t n 0) ω ≤ C) :
Tendsto (fun n => ∑ i ∈ Finset.range (N n),
∫ ω, (A (t n (i + 1)) ω - A (t n i) ω) ^ 2 ∂μ) atTop (𝓝 0) := by
have heq (n : ℕ) : (∑ i ∈ Finset.range (N n),
∫ ω, (A (t n (i + 1)) ω - A (t n i) ω) ^ 2 ∂μ) =
∫ ω, squaredIncrements (fun i => A (t n i) ω) (N n) ∂μ := by
symm
apply integral_finsetSum
intro i hi
apply (memLp_increment_of_total_bound (fun j => hmeas (t n j))
(hpos.mono (fun ω hω => hω n))
(hbound.mono (fun ω hω => hω n)) (Finset.mem_range.mp hi) 2).integrable_sq
simp_rw [heq]
exact tendsto_integral_squaredIncrements hK hmeas hc t N mesh hmem hdist hmesh hpos hC hbound
end Problem367.CompensatorVariation
end
end OAI
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