File size: 2,017 Bytes
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 | import Mathlib
import OAI.Probability.Ballisticity.Entropy.EntropyVariational
namespace OAI
section
open MeasureTheory ProbabilityTheory InformationTheory Filter
open scoped ENNReal NNReal Classical Topology
namespace DirectionalTransience.Entropy
variable {X : Type*} [MeasurableSpace X]
lemma event_budget (μ ν : Measure X) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
{C M t : ℝ} (hC : 0≤C) (hkl : klDiv μ ν≤ENNReal.ofReal C)
(S : Set X) (hS : MeasurableSet S) (he : Real.exp t * ν.real S≤M) :
μ.real S*t≤C+M := by
let f : X → ℝ := S.indicator (fun _ => t)
have hf : Measurable f := measurable_const.indicator hS
have hb : ∃ B : ℝ, ∀ x, |f x|≤B := ⟨|t|,fun x => by
by_cases hx : x∈S <;> simp [f,hx]⟩
have hi : (∫ x, f x ∂μ)=μ.real S*t := by
simp only [f,integral_indicator_const t hS,smul_eq_mul]
have heq : (fun x => Real.exp (f x))=fun x => S.indicator (fun _ => Real.exp t-1) x+1 := by
funext x
by_cases hx : x∈S <;> simp [f,hx]
have hh := expBudget_of_kl_le μ ν hC hkl f hf hb
rw [hi,heq,integral_add ((integrable_const _).indicator hS) (integrable_const _),
integral_indicator_const _ hS,integral_const,probReal_univ,one_smul,smul_eq_mul] at hh
have hn : 0≤ν.real S := measureReal_nonneg
nlinarith
lemma event_moment_budget (μ ν : Measure X) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
{C : ℝ} (hC : 0≤C) (hkl : klDiv μ ν≤ENNReal.ofReal C)
(F : X → ℝ≥0∞) (hF : Measurable F) {M : ℝ≥0∞} (hM : M≠⊤)
(hmean : (∫⁻ x, F x ∂ν)≤M) (t : ℝ) :
μ.real {x | ENNReal.ofReal (Real.exp t)≤F x}*t ≤ C+M.toReal := by
have hm := (mul_meas_ge_le_lintegral hF (ENNReal.ofReal (Real.exp t))).trans hmean
have hr := ENNReal.toReal_mono hM hm
simp only [ENNReal.toReal_mul,ENNReal.toReal_ofReal (Real.exp_nonneg _)] at hr
exact event_budget μ ν hC hkl _ (measurableSet_le measurable_const hF) hr
end DirectionalTransience.Entropy
end
end OAI
|