| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory Filter |
| open scoped BigOperators |
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| namespace Problem367.DyadicIncrementControl |
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| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] |
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| 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 |
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| 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) |
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| 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⟩) |
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| 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 |
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| 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 |
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| end Problem367.DyadicIncrementControl |
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| end |
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| end OAI |
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