| import OAI.Analysis.IrrationalRotation.Model.Analytic |
| import OAI.Analysis.IrrationalRotation.Rotation.ExponentialWeight |
| import OAI.Analysis.IrrationalRotation.Analysis.TruncatedLogIntegral |
|
|
| namespace OAI |
|
|
| open Filter MeasureTheory |
| open scoped ENNReal Topology |
|
|
| namespace Invariant233 |
|
|
| @[simp] theorem negLogCost_zero : negLogCost 0 = ∞ := by |
| simp [negLogCost] |
|
|
| theorem negLogCost_of_pos {t : ℝ} (ht : 0 < t) : |
| negLogCost t = ENNReal.ofReal (-Real.log t) := by |
| simp [negLogCost, ht.ne', abs_of_pos ht] |
|
|
| variable {X : Type*} [MeasurableSpace X] {μ : Measure X} |
|
|
| theorem lintegral_negLogCost_eq_of_ae_pos (f : X → ℝ) |
| (hpos : ∀ᵐ x ∂μ, 0 < f x) : |
| (∫⁻ x, negLogCost (f x) ∂μ) = |
| ∫⁻ x, ENNReal.ofReal (-Real.log (f x)) ∂μ := by |
| apply lintegral_congr_ae |
| filter_upwards [hpos] with x hx |
| exact negLogCost_of_pos hx |
|
|
| theorem hasNegInfiniteLogIntegral_iff_of_ae_pos (f : X → ℝ) |
| (hpos : ∀ᵐ x ∂μ, 0 < f x) : |
| HasNegInfiniteLogIntegral μ f ↔ |
| (∫⁻ x, ENNReal.ofReal (-Real.log (f x)) ∂μ) = ∞ := by |
| unfold HasNegInfiniteLogIntegral |
| rw [lintegral_negLogCost_eq_of_ae_pos f hpos] |
|
|
| theorem scalarWeightDeterminant_eq_zero_of_log_integral |
| (f : X → ℝ) (h : HasNegInfiniteLogIntegral μ f) : |
| scalarWeightDeterminant μ f = 0 := by |
| unfold scalarWeightDeterminant |
| rw [show (∫⁻ x, negLogCost (f x) ∂μ) = ∞ from h] |
| simp |
|
|
| theorem ae_pos_of_nonneg_singleton_zero (f : X → ℝ) (a : X) |
| (ha : μ {a} = 0) (hf : ∀ x, 0 ≤ f x) |
| (hz : ∀ x, f x = 0 ↔ x = a) : ∀ᵐ x ∂μ, 0 < f x := by |
| have hne : ∀ᵐ x ∂μ, x ≠ a := by |
| simpa only [ae_iff, not_not, Set.ofPred_eq_eq_singleton] using ha |
| filter_upwards [hne] with x hx |
| exact lt_of_le_of_ne (hf x) (Ne.symm (fun h => hx ((hz x).mp h))) |
|
|
| @[simp] theorem circleMeasure_singleton (x : Circle) : circleMeasure {x} = 0 := by |
| change volume {x} = 0 |
| rw [← Metric.closedBall_zero, AddCircle.volume_closedBall] |
| norm_num |
|
|
| theorem IsContinuousCircleWeight.ae_pos {f : Circle → ℝ} |
| (hf : IsContinuousCircleWeight f) : ∀ᵐ x ∂circleMeasure, 0 < f x := by |
| apply ae_pos_of_nonneg_singleton_zero f 0 |
| · simp only [circleMeasure_singleton] |
| · exact fun x => (hf.2.1 x).1 |
| · exact hf.2.2.1 |
|
|
| theorem exists_continuous_circle_weight_of_cost (g : Circle → ℝ) |
| (hc : ContinuousOn g ({0}ᶜ)) |
| (hg : ∀ x ≠ 0, 0 ≤ g x) |
| (hlim : Tendsto g (𝓝[≠] 0) atTop) |
| (hint : (∫⁻ x, ENNReal.ofReal (g x) ∂circleMeasure) = ∞) : |
| ∃ f : Circle → ℝ, IsContinuousCircleWeight f ∧ |
| (∀ x ≠ 0, -Real.log (f x) = g x) := by |
| obtain ⟨f, hfcont, hfrange, hfzero, hflog, hfinf⟩ := |
| exists_continuous_weight_of_cost circleMeasure 0 g (by simp only [circleMeasure_singleton]) |
| hc hg hlim hint |
| have hpos : ∀ᵐ x ∂circleMeasure, 0 < f x := |
| ae_pos_of_nonneg_singleton_zero f 0 (by simp only [circleMeasure_singleton]) |
| (fun x => (hfrange x).1) hfzero |
| exact ⟨f, ⟨hfcont, hfrange, hfzero, |
| (hasNegInfiniteLogIntegral_iff_of_ae_pos f hpos).mpr hfinf⟩, hflog⟩ |
|
|
| theorem IsContinuousCircleWeight.exists_truncated_log_integral_lt |
| {f : Circle → ℝ} (hf : IsContinuousCircleWeight f) (a : ℝ) : |
| ∃ M : ℝ, 0 < M ∧ |
| (∫ x, Real.log (max (f x) (Real.exp (-M))) ∂circleMeasure) < -2 * a := by |
| let : IsFiniteMeasure circleMeasure := by unfold circleMeasure; infer_instance |
| apply exists_pos_truncated_log_integral_lt f hf.1.measurable |
| (fun x => (hf.2.1 x).1) (fun x => (hf.2.1 x).2) hf.ae_pos |
| exact (hasNegInfiniteLogIntegral_iff_of_ae_pos f hf.ae_pos).mp hf.2.2.2 |
|
|
| end Invariant233 |
|
|
| end OAI |
|
|
