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f61e547 | 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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped BigOperators
namespace Problem367.DyadicIncrementControl
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ]
theorem measureReal_abs_gt_le_fourth {X : Ω → ℝ} {a : ℝ} (ha : 0 < a)
(hX : Integrable (fun ω => |X ω| ^ 4) μ) :
μ.real {ω | a < |X ω|} ≤ (∫ ω, |X ω| ^ 4 ∂μ) / a ^ 4 := by
have hm := mul_meas_ge_le_integral_of_nonneg
(Filter.Eventually.of_forall (fun ω => pow_nonneg (abs_nonneg (X ω)) 4)) hX (a ^ 4)
apply (le_div_iff₀ (pow_pos ha 4)).2
calc
μ.real {ω | a < |X ω|} * a ^ 4 ≤
μ.real {ω | a ^ 4 ≤ |X ω| ^ 4} * a ^ 4 := by
apply mul_le_mul_of_nonneg_right _ (pow_nonneg ha.le 4)
exact measureReal_mono (fun ω hω => pow_le_pow_left₀ ha.le hω.le 4)
_ ≤ ∫ ω, |X ω| ^ 4 ∂μ := by simpa only [mul_comm] using hm
theorem ae_eventually_of_measureReal_le {E : ℕ → Set Ω} {b : ℕ → ℝ}
(hb : Summable b) (hE : ∀ n, μ.real (E n) ≤ b n) :
∀ᵐ ω ∂μ, ∀ᶠ n in atTop, ω ∉ E n := by
apply ae_eventually_notMem
apply ne_top_of_le_ne_top hb.tsum_ofReal_ne_top
apply ENNReal.tsum_le_tsum
intro n
rw [← ENNReal.ofReal_toReal (measure_ne_top μ (E n))]
exact ENNReal.ofReal_le_ofReal (hE n)
theorem ae_eventually_grid_bound {m : ℕ → ℕ}
(X : ∀ n, Fin (m n) → Ω → ℝ) {C q b : ℝ}
(hq : 0 ≤ q) (hb : 0 < b) (hqb : q < b ^ 4)
(hInt : ∀ n i, Integrable (fun ω => |X n i ω| ^ 4) μ)
(hMoment : ∀ n, (∑ i : Fin (m n), ∫ ω, |X n i ω| ^ 4 ∂μ) ≤ C * q ^ n) :
∀ᵐ ω ∂μ, ∀ᶠ n in atTop, ∀ i : Fin (m n), |X n i ω| ≤ b ^ n := by
classical
let E : ℕ → Set Ω := fun n => {ω | ∃ i : Fin (m n), b ^ n < |X n i ω|}
have hratio0 : 0 ≤ q / b ^ 4 := div_nonneg hq (pow_nonneg hb.le _)
have hratio1 : q / b ^ 4 < 1 := (div_lt_one (pow_pos hb 4)).2 hqb
have hsum : Summable (fun n : ℕ => C * (q / b ^ 4) ^ n) :=
(summable_geometric_of_lt_one hratio0 hratio1).mul_left C
have hbound (n : ℕ) : μ.real (E n) ≤ C * (q / b ^ 4) ^ n := by
calc
μ.real (E n) = μ.real (⋃ i : Fin (m n), {ω | b ^ n < |X n i ω|}) := by
congr 1
ext ω
simp [E]
_ ≤ ∑ i : Fin (m n), μ.real {ω | b ^ n < |X n i ω|} :=
measureReal_iUnion_fintype_le _
_ ≤ ∑ i : Fin (m n), (∫ ω, |X n i ω| ^ 4 ∂μ) / (b ^ n) ^ 4 :=
Finset.sum_le_sum (fun i _ => measureReal_abs_gt_le_fourth (pow_pos hb n) (hInt n i))
_ = (∑ i : Fin (m n), ∫ ω, |X n i ω| ^ 4 ∂μ) / (b ^ n) ^ 4 := by
rw [Finset.sum_div]
_ ≤ (C * q ^ n) / (b ^ n) ^ 4 :=
div_le_div_of_nonneg_right (hMoment n) (pow_nonneg (pow_nonneg hb.le n) 4)
_ = C * (q / b ^ 4) ^ n := by
rw [div_pow, ← pow_mul, ← pow_mul]
rw [Nat.mul_comm n 4]
ring
filter_upwards [ae_eventually_of_measureReal_le hsum hbound] with ω hω
filter_upwards [hω] with n hn i
exact le_of_not_gt (fun hi => hn ⟨i, hi⟩)
theorem ae_eventually_dyadic_grid_bound (K : ℕ)
(X : ∀ n, Fin (K * 2 ^ n) → Ω → ℝ) {C b : ℝ}
(hb : 0 < b) (hdecay : (1 / 2 : ℝ) < b ^ 4)
(hInt : ∀ n i, Integrable (fun ω => |X n i ω| ^ 4) μ)
(hMoment : ∀ n i, (∫ ω, |X n i ω| ^ 4 ∂μ) ≤ C * (1 / 4 : ℝ) ^ n) :
∀ᵐ ω ∂μ, ∀ᶠ n in atTop, ∀ i : Fin (K * 2 ^ n), |X n i ω| ≤ b ^ n := by
apply ae_eventually_grid_bound X (C := (K : ℝ) * C) (q := 1 / 2)
(by norm_num) hb hdecay hInt
intro n
calc
(∑ i : Fin (K * 2 ^ n), ∫ ω, |X n i ω| ^ 4 ∂μ) ≤
∑ _i : Fin (K * 2 ^ n), C * (1 / 4 : ℝ) ^ n :=
Finset.sum_le_sum (fun i _ => hMoment n i)
_ = ((K : ℝ) * C) * (1 / 2 : ℝ) ^ n := by
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin,
nsmul_eq_mul, Nat.cast_mul, Nat.cast_pow, Nat.cast_ofNat]
calc
((K : ℝ) * 2 ^ n) * (C * (1 / 4 : ℝ) ^ n) =
((K : ℝ) * C) * ((2 : ℝ) ^ n * (1 / 4 : ℝ) ^ n) := by ring
_ = ((K : ℝ) * C) * (1 / 2 : ℝ) ^ n := by
rw [← mul_pow]
norm_num
theorem ae_eventually_dyadic_grid_nine_tenths (K : ℕ)
(X : ∀ n, Fin (K * 2 ^ n) → Ω → ℝ) {C : ℝ}
(hInt : ∀ n i, Integrable (fun ω => |X n i ω| ^ 4) μ)
(hMoment : ∀ n i, (∫ ω, |X n i ω| ^ 4 ∂μ) ≤ C * (1 / 4 : ℝ) ^ n) :
∀ᵐ ω ∂μ, ∀ᶠ n in atTop, ∀ i : Fin (K * 2 ^ n),
|X n i ω| ≤ (9 / 10 : ℝ) ^ n :=
ae_eventually_dyadic_grid_bound K X (by norm_num) (by norm_num) hInt hMoment
end Problem367.DyadicIncrementControl
end
end OAI
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