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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 | 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
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