| import Mathlib |
| import OAI.Probability.Ballisticity.Entropy.EntropyPlus |
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| namespace OAI |
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| section |
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| open MeasureTheory ProbabilityTheory InformationTheory |
| open scoped ENNReal |
| namespace DirectionalTransience.Entropy |
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|
| lemma density_of_finite_kl {X : Type*} [MeasurableSpace X] |
| (μ ν : Measure X) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| (hkl : klDiv μ ν≠∞) : |
| ∃ g : X → ℝ, Measurable g ∧ (∀ x, 0≤g x) ∧ Integrable g ν ∧ |
| (∫ x, g x ∂ν)=1 ∧ Integrable (fun x => TailDecorrelation.entropyPlus (g x)) ν ∧ |
| μ=ν.withDensity (fun x => ENNReal.ofReal (g x)) := by |
| obtain ⟨hac,hint⟩ := klDiv_ne_top_iff.mp hkl |
| let g : X → ℝ := fun x => (μ.rnDeriv ν x).toReal |
| have hg : Measurable g := (Measure.measurable_rnDeriv μ ν).ennreal_toReal |
| have hgi : Integrable g ν := by |
| simpa only [mul_one] using |
| (integrable_toReal_rnDeriv_mul_iff hac (f := fun _ => (1:ℝ))).mpr (integrable_const 1) |
| have hg1 : (∫ x, g x ∂ν)=1 := by |
| simpa only [mul_one,integral_const,probReal_univ,one_smul] using |
| (integral_toReal_rnDeriv_mul hac (f := fun _ => (1:ℝ))) |
| have hpos : Integrable (fun x => Real.log (max 1 (g x))) μ := by |
| apply hint.abs.mono' (measurable_const.max hg).log.aestronglyMeasurable |
| apply ae_of_all |
| intro x |
| rw [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg (le_max_left _ _))] |
| change Real.log (max 1 (g x))≤|Real.log (g x)| |
| by_cases h : g x≤1 |
| · simp only [max_eq_left h,Real.log_one,abs_nonneg] |
| · rw [max_eq_right (le_of_not_ge h)] |
| exact le_abs_self _ |
| have hH := (integrable_toReal_rnDeriv_mul_iff hac).mpr hpos |
| refine ⟨g,hg,fun _ => ENNReal.toReal_nonneg,hgi,hg1,hH,?_⟩ |
| rw [←Measure.withDensity_rnDeriv_eq μ ν hac] |
| apply withDensity_congr_ae |
| filter_upwards [Measure.rnDeriv_ne_top μ ν] with x hx |
| exact (ENNReal.ofReal_toReal hx).symm |
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| end DirectionalTransience.Entropy |
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| end |
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| end OAI |
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