| import Mathlib |
| import OAI.Probability.GaussianInformation.Dyadic.SmallBalls |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
| open MeasureTheory Metric Set |
| open scoped ENNReal NNReal |
| open MeasureTheory Set Filter |
| open scoped ENNReal Topology |
| namespace RepeatedLocalization |
| open Metric |
|
|
| noncomputable def tubeWeightConstant (n : ℕ) (B : ℝ) : ℝ≥0∞ := |
| ENNReal.ofReal (B * tubeConstant^n) ^ (1/2 : ℝ) |
|
|
| lemma tubeWeightConstant_ne_top (n : ℕ) (B : ℝ) : tubeWeightConstant n B ≠ ∞ := by |
| exact ENNReal.rpow_ne_top_of_nonneg (by norm_num) ENNReal.ofReal_ne_top |
|
|
| lemma weighted_dyadic_tube_tail {n : ℕ} (hn : 0 < n) (Q : Cell n) |
| (μ : Measure (Coordinate n)) (hsupport : μ (cellSet Q)ᶜ = 0) |
| {B : ℝ} (hB : 0 ≤ B) (hspread : DyadicSpread μ Q B) |
| (h : Coordinate n → ℝ) (hh : Measurable h) (hnorm : eLpNorm h 2 μ ≤ 1) |
| (V : AffineSubspace ℝ (EuclideanSpace ℝ (Fin (n+1)))) |
| (hV : (V : Set (EuclideanSpace ℝ (Fin (n+1)))).Nonempty) |
| (k : ℕ) (he : Module.finrank ℝ V.direction + 2*(k+1) ≤ n/2) (j : ℕ) : |
| (μ.withDensity (fun z => ENNReal.ofReal |h z|)) |
| {z | infDist (normalizedChart Q z) (V : Set (EuclideanSpace ℝ (Fin (n+1)))) ≤ (1/2:ℝ)^j} |
| ≤ tubeWeightConstant n B * ENNReal.ofReal ((1/2:ℝ)^((k+1)*j)) := by |
| have hmeas := (continuous_infDist_pt (V : Set (EuclideanSpace ℝ (Fin (n+1))))).measurable.comp |
| (normalizedChart_measurable Q) |
| refine (weightedMeasure_apply_le_sqrt μ h hh hnorm _ |
| (measurableSet_le hmeas measurable_const)).trans ?_ |
| refine (ENNReal.rpow_le_rpow (dyadic_tube_tail hn Q μ hsupport hB hspread V hV _ he j) |
| (by norm_num : (0:ℝ) ≤ 1/2)).trans_eq ?_ |
| rw [ENNReal.ofReal_mul (mul_nonneg hB (pow_nonneg (le_trans (by norm_num) tubeConstant_ge_one) _)), |
| ENNReal.mul_rpow_of_nonneg _ _ (by norm_num : (0:ℝ) ≤ 1/2)] |
| congr 1 |
| have hp : (1/2:ℝ)^(2*(k+1)*j) = ((1/2:ℝ)^((k+1)*j))^2 := by |
| rw [← pow_mul] |
| congr 1 |
| ring |
| rw [hp, ENNReal.ofReal_pow (by positivity), ← ENNReal.rpow_natCast, |
| ← ENNReal.rpow_mul] |
| norm_num |
|
|
| lemma weighted_affine_inverseMoment {n : ℕ} (hn : 0 < n) (Q : Cell n) |
| (μ : Measure (Coordinate n)) [IsProbabilityMeasure μ] |
| (hsupport : μ (cellSet Q)ᶜ = 0) {B : ℝ} (hB : 0 ≤ B) (hspread : DyadicSpread μ Q B) |
| (h : Coordinate n → ℝ) (hh : Measurable h) (hnorm : eLpNorm h 2 μ ≤ 1) |
| (V : AffineSubspace ℝ (EuclideanSpace ℝ (Fin (n+1)))) |
| (hV : (V : Set (EuclideanSpace ℝ (Fin (n+1)))).Nonempty) |
| (k : ℕ) (hk : 0 < k) (he : Module.finrank ℝ V.direction + 2*(k+1) ≤ n/2) : |
| (∫⁻ z, ENNReal.ofReal ((infDist (normalizedChart Q z) |
| (V : Set (EuclideanSpace ℝ (Fin (n+1)))))⁻¹ ^ k) |
| ∂(μ.withDensity (fun z => ENNReal.ofReal |h z|))) ≤ |
| 1 + tubeWeightConstant n B * ENNReal.ofReal ((2:ℝ)^(k+1)) := by |
| refine (inverseMoment_of_dyadic_tails _ _ |
| ((continuous_infDist_pt _).measurable.comp (normalizedChart_measurable Q)) |
| (fun _ => infDist_nonneg) k hk _ |
| (weighted_dyadic_tube_tail hn Q μ hsupport hB hspread h hh hnorm V hV k he)).trans ?_ |
| gcongr |
| exact weightedMeasure_mass_le_one μ h hh hnorm |
|
|
| lemma weighted_affine_zero_null {n : ℕ} (hn : 0 < n) (Q : Cell n) |
| (μ : Measure (Coordinate n)) (hsupport : μ (cellSet Q)ᶜ = 0) |
| {B : ℝ} (hB : 0 ≤ B) (hspread : DyadicSpread μ Q B) |
| (h : Coordinate n → ℝ) (hh : Measurable h) (hnorm : eLpNorm h 2 μ ≤ 1) |
| (V : AffineSubspace ℝ (EuclideanSpace ℝ (Fin (n+1)))) |
| (hV : (V : Set (EuclideanSpace ℝ (Fin (n+1)))).Nonempty) |
| (k : ℕ) (he : Module.finrank ℝ V.direction + 2*(k+1) ≤ n/2) : |
| (μ.withDensity (fun z => ENNReal.ofReal |h z|)) |
| {z | infDist (normalizedChart Q z) (V : Set (EuclideanSpace ℝ (Fin (n+1)))) = 0} = 0 := by |
| exact zeroSet_null_of_dyadic_tails _ _ k _ (tubeWeightConstant_ne_top n B) |
| (weighted_dyadic_tube_tail hn Q μ hsupport hB hspread h hh hnorm V hV k he) |
|
|
| lemma weighted_affine_inverseMoment_ennreal {n : ℕ} (hn : 0 < n) (Q : Cell n) |
| (μ : Measure (Coordinate n)) [IsProbabilityMeasure μ] |
| (hsupport : μ (cellSet Q)ᶜ = 0) {B : ℝ} (hB : 0 ≤ B) (hspread : DyadicSpread μ Q B) |
| (h : Coordinate n → ℝ) (hh : Measurable h) (hnorm : eLpNorm h 2 μ ≤ 1) |
| (V : AffineSubspace ℝ (EuclideanSpace ℝ (Fin (n+1)))) |
| (hV : (V : Set (EuclideanSpace ℝ (Fin (n+1)))).Nonempty) |
| (k : ℕ) (hk : 0 < k) (he : Module.finrank ℝ V.direction + 2*(k+1) ≤ n/2) : |
| (∫⁻ z, (ENNReal.ofReal (infDist (normalizedChart Q z) |
| (V : Set (EuclideanSpace ℝ (Fin (n+1))))))⁻¹ ^ k |
| ∂(μ.withDensity (fun z => ENNReal.ofReal |h z|))) ≤ |
| 1 + tubeWeightConstant n B * ENNReal.ofReal ((2:ℝ)^(k+1)) := by |
| have hzero : ∀ᵐ z ∂(μ.withDensity (fun z => ENNReal.ofReal |h z|)), |
| infDist (normalizedChart Q z) (V : Set (EuclideanSpace ℝ (Fin (n+1)))) ≠ 0 := by |
| apply ae_iff.mpr |
| simpa only [not_not] using weighted_affine_zero_null hn Q μ hsupport hB hspread h hh hnorm V hV k he |
| have heq : (fun z => (ENNReal.ofReal (infDist (normalizedChart Q z) |
| (V : Set (EuclideanSpace ℝ (Fin (n+1))))))⁻¹ ^ k) =ᵐ[μ.withDensity (fun z => ENNReal.ofReal |h z|)] |
| (fun z => ENNReal.ofReal ((infDist (normalizedChart Q z) |
| (V : Set (EuclideanSpace ℝ (Fin (n+1)))))⁻¹ ^ k)) := by |
| filter_upwards [hzero] with z hz |
| rw [ENNReal.ofReal_pow (inv_nonneg.mpr infDist_nonneg), |
| ENNReal.ofReal_inv_of_pos (lt_of_le_of_ne infDist_nonneg hz.symm)] |
| rw [lintegral_congr_ae heq] |
| exact weighted_affine_inverseMoment hn Q μ hsupport hB hspread h hh hnorm V hV k hk he |
|
|
| end RepeatedLocalization |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
|
|
| end OAI |
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