| import Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp |
| import Mathlib.Analysis.SpecialFunctions.Pow.Real |
| import Mathlib.MeasureTheory.Group.AddCircle |
| import Mathlib.MeasureTheory.Integral.Lebesgue.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Filter MeasureTheory |
| open scoped ENNReal Topology |
|
|
| namespace Invariant233 |
|
|
| abbrev Circle := AddCircle (1 : ℝ) |
|
|
| def circleMeasure : Measure Circle := volume |
|
|
| def negLogCost (t : ℝ) : ℝ≥0∞ := |
| if t = 0 then ∞ else ENNReal.ofReal (-Real.log |t|) |
|
|
| def HasNegInfiniteLogIntegral {X : Type*} [MeasurableSpace X] |
| (μ : Measure X) (f : X → ℝ) : Prop := |
| (∫⁻ x, negLogCost (f x) ∂μ) = ∞ |
|
|
| def scalarWeightDeterminant {X : Type*} [MeasurableSpace X] |
| (μ : Measure X) (f : X → ℝ) : ℝ≥0∞ := |
| EReal.exp (-((∫⁻ x, negLogCost (f x) ∂μ) : EReal)) |
|
|
| def NormQuasinilpotent {A : Type*} [NormedRing A] (T : A) : Prop := |
| Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) |
|
|
| def IsContinuousCircleWeight (f : Circle → ℝ) : Prop := |
| Continuous f ∧ (∀ x, 0 ≤ f x ∧ f x ≤ 1) ∧ |
| (∀ x, f x = 0 ↔ x = 0) ∧ HasNegInfiniteLogIntegral circleMeasure f |
|
|
| end Invariant233 |
|
|
| end |
|
|
| end OAI |
|
|
