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9eeb9c7 | 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 | 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
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