| import Mathlib |
| import OAI.Probability.Ballisticity.Entropy.EntropyVariational |
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| namespace OAI |
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| section |
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| open MeasureTheory ProbabilityTheory InformationTheory Filter |
| open scoped ENNReal NNReal Classical Topology |
| namespace DirectionalTransience.Entropy |
| variable {X : Type*} [MeasurableSpace X] |
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| 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 |
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| 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 |
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| end DirectionalTransience.Entropy |
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| end |
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| end OAI |
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