| import OAI.MeasureTheory.SelfSimilar.BallIntegrability |
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| namespace OAI |
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| universe uι |
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| section |
| open scoped BigOperators ENNReal |
| open MeasureTheory |
| open MeasureTheory Metric Set Filter |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set ProbabilityTheory |
| open scoped Topology |
| open MeasureTheory Metric Set Filter |
| open scoped ENNReal Topology |
| namespace EntropyRateDimension.Coding |
| open MeasureTheory Metric Set Filter EntropyRateDimension.Bernoulli |
| open scoped Topology BigOperators ENNReal |
| variable {ι : Type uι} [Fintype ι] [Nonempty ι] |
| local instance : MeasurableSpace ι := ⊤ |
| local instance : MeasurableSingletonClass ι := ⟨fun _ => trivial⟩ |
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| theorem head_point_measurable (S : System ι) : |
| Measurable (fun ω : ℕ → ι => (ω 0,point S ω)) := |
| (measurable_pi_apply 0).prodMk (point_measurable S) |
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| theorem head_point_law (S : System ι) : |
| (addressMeasure S).map (fun ω => (ω 0,point S ω)) = |
| ∑ i, (Measure.dirac i).prod (S.component (law S) i) := by |
| classical |
| let f : ι × ℝ → ι × ℝ := fun z => (z.1,S.affine z.1 z.2) |
| have hf : Measurable f := by |
| apply measurable_fst.prodMk |
| exact (((measurable_of_finite S.ratio).comp measurable_fst).mul measurable_snd).add |
| ((measurable_of_finite S.offset).comp measurable_fst) |
| have hh := congrArg (Measure.map f) (head_tail_point_law S) |
| have hp : Measurable (fun ω : ℕ → ι => (ω 0,point S (shift ω))) := |
| (measurable_pi_apply 0).prodMk ((point_measurable S).comp shift_measurable) |
| rw [Measure.map_map hf hp] at hh |
| have he : f ∘ (fun ω => (ω 0,point S (shift ω))) = fun ω => (ω 0,point S ω) := by |
| funext ω |
| exact Prod.ext rfl (point_shift S ω).symm |
| rw [he] at hh |
| rw [hh] |
| change ((∑ i, ENNReal.ofReal (S.weight i) • Measure.dirac i).prod (law S)).map f = _ |
| rw [← Measure.sum_fintype,Measure.prod_sum_left,Measure.sum_fintype, |
| Measure.map_finset_sum hf.aemeasurable] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [Measure.prod_smul_left,Measure.map_smul _ hf.aemeasurable,Measure.dirac_prod, |
| Measure.map_map hf measurable_prodMk_left,System.component,Measure.prod_smul_right, |
| Measure.dirac_prod,Measure.map_map measurable_prodMk_left (S.affine_continuous i).measurable] |
| rfl |
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| theorem ae_head_point (S : System ι) {P : ι → ℝ → Prop} |
| (hP : ∀ i, ∀ᵐ x ∂S.component (law S) i, P i x) : |
| ∀ᵐ ω ∂addressMeasure S, P (ω 0) (point S ω) := by |
| have hh : ∀ᵐ z ∂(addressMeasure S).map (fun ω => (ω 0,point S ω)), P z.1 z.2 := by |
| rw [head_point_law,ae_finsetSum_measure_iff] |
| intro i _ |
| rw [Measure.dirac_prod,(measurableEmbedding_prodMk_left i).ae_map_iff] |
| exact hP i |
| exact ae_of_ae_map (head_point_measurable S).aemeasurable hh |
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| theorem integrable_head_point (S : System ι) {f : ι → ℝ → ℝ} |
| (hfm : ∀ i, Measurable (f i)) (hf : ∀ i, Integrable (f i) (S.component (law S) i)) : |
| Integrable (fun ω => f (ω 0) (point S ω)) (addressMeasure S) := by |
| have hm : Measurable (fun z : ι × ℝ => f z.1 z.2) := |
| measurable_from_prod_countable_right hfm |
| have hi : Integrable (fun z : ι × ℝ => f z.1 z.2) |
| ((addressMeasure S).map (fun ω => (ω 0,point S ω))) := by |
| rw [head_point_law,integrable_finsetSum_measure] |
| intro i _ |
| rw [Measure.dirac_prod,integrable_map_measure hm.aestronglyMeasurable |
| measurable_prodMk_left.aemeasurable] |
| exact hf i |
| exact hi.comp_measurable (head_point_measurable S) |
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| noncomputable def envelope (S : System ι) (ω : ℕ → ι) : ℝ := |
| BallRatios.logEnvelope (law S) (S.component (law S) (ω 0)) (point S ω) |
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| theorem envelope_joint_measurable (S : System ι) : Measurable (fun z : ι × ℝ => |
| BallRatios.logEnvelope (law S) (S.component (law S) z.1) z.2) := by |
| apply measurable_from_prod_countable_right |
| intro i |
| exact BallRatios.logEnvelope_measurable (law S) (S.component (law S) i) |
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| theorem envelope_measurable (S : System ι) : Measurable (envelope S) := by |
| unfold envelope |
| have hh := (envelope_joint_measurable S).comp (head_point_measurable S) |
| exact hh |
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| omit [Nonempty ι] in |
| theorem envelope_nonneg (S : System ι) (ω : ℕ → ι) : 0 ≤ envelope S ω := |
| BallRatios.logEnvelope_nonneg _ _ _ |
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| theorem envelope_integrable (S : System ι) : Integrable (envelope S) (addressMeasure S) := |
| integrable_head_point S (fun i => BallRatios.logEnvelope_measurable _ (S.component (law S) i)) |
| (fun i => BallRatios.logEnvelope_integrable _ (S.component (law S) i)) |
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| noncomputable def corr (S : System ι) (n : ℕ) (ω : ℕ → ι) : ℝ := |
| BallRatios.logRatio (law S) (S.component (law S) (ω 0)) (point S ω) |slope S ω n| |
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| noncomputable def corrLimit (S : System ι) (ω : ℕ → ι) : ℝ := |
| Real.log ((law S).rnDeriv (S.component (law S) (ω 0)) (point S ω)).toReal |
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| theorem corr_joint_measurable (S : System ι) : Measurable (fun z : ι × (ℝ × ℝ) => |
| BallRatios.logRatio (law S) (S.component (law S) z.1) z.2.1 z.2.2) := by |
| apply measurable_from_prod_countable_right |
| intro i |
| exact BallRatios.logRatio_measurable (law S) (S.component (law S) i) |
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| theorem corr_measurable (S : System ι) (n : ℕ) : Measurable (corr S n) := by |
| unfold corr |
| have hh := (corr_joint_measurable S).comp ((measurable_pi_apply 0).prodMk |
| ((point_measurable S).prodMk (slope_measurable S n).abs)) |
| exact hh |
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| theorem corrLimit_measurable (S : System ι) : Measurable (corrLimit S) := by |
| have hm : Measurable (fun z : ι × ℝ => |
| Real.log ((law S).rnDeriv (S.component (law S) z.1) z.2).toReal) := by |
| apply measurable_from_prod_countable_right |
| intro i |
| exact ((law S).measurable_rnDeriv (S.component (law S) i)).ennreal_toReal.log |
| exact hm.comp (head_point_measurable S) |
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| theorem ae_corr_bounds (S : System ι) : ∀ᵐ ω ∂addressMeasure S, ∀ n : ℕ, 0 < n → |
| 0 ≤ corr S n ω ∧ corr S n ω ≤ envelope S ω := by |
| filter_upwards [ae_head_point S (fun i => BallRatios.logRatio_bounds _ _ |
| (S.component_le _ (law_selfSimilar S) i))] with ω hω |
| intro n hn |
| exact hω _ (abs_slope_pos S ω n) (abs_slope_lt_one S ω hn) |
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| theorem ae_corr_limit (S : System ι) : ∀ᵐ ω ∂addressMeasure S, |
| Tendsto (fun n => corr S n ω) atTop (𝓝 (corrLimit S ω)) := by |
| filter_upwards [ae_head_point S (fun i => BallRatios.ae_logRatio_limit _ _ |
| (S.component_le _ (law_selfSimilar S) i))] with ω hω |
| apply hω.2.comp |
| exact tendsto_nhdsWithin_iff.mpr ⟨slope_tendsto_zero S ω, |
| Eventually.of_forall (abs_slope_pos S ω)⟩ |
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| end EntropyRateDimension.Coding |
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| end |
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| end OAI |
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