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180bc03 | 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 | import Mathlib
import OAI.Probability.Ballisticity.Entropy.EntropyPlus
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory
open scoped ENNReal
namespace DirectionalTransience.Entropy
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
end DirectionalTransience.Entropy
end
end OAI
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