| import OAI.NumberTheory.DukePrimeDegree.Entropy.ConditionalSigmaOrder |
|
|
| namespace OAI |
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| /-! # Entropy along increasing conditioning sigma algebras |
|
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| The martingale limit applies also when the conditioning includes an |
| arbitrary transverse coordinate in addition to finite future names. |
| -/ |
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|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped BigOperators Topology |
| namespace DukePrimeDegree |
| open PrimeDegreePackets |
|
|
| lemma finiteShannon_norm_le_card_of_unit_bounds {α : Type*} [Fintype α] |
| (p : α → ℝ) (hp : ∀ a, p a ∈ Icc 0 1) : |
| ‖finiteShannon p‖ ≤ Fintype.card α := by |
| unfold finiteShannon |
| calc |
| ‖∑ a, Real.negMulLog (p a)‖ ≤ ∑ a, ‖Real.negMulLog (p a)‖ := norm_sum_le _ _ |
| _ ≤ ∑ _a : α, (1:ℝ) := by |
| apply Finset.sum_le_sum |
| intro a _ |
| rw [Real.norm_eq_abs,abs_of_nonneg (Real.negMulLog_nonneg (hp a).1 (hp a).2)] |
| have H := Real.negMulLog_le_one_sub_self (hp a).1 |
| linarith [(hp a).1] |
| _ = _ := by simp |
|
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| theorem conditionalSigmaEntropy_tendsto_iSup {Y α : Type*} |
| [mY : MeasurableSpace Y] [Fintype α] |
| (μ : Measure Y) [IsFiniteMeasure μ] (P : Y → α) |
| (m : ℕ → MeasurableSpace Y) (hm : Monotone m) (hle : ∀ n, m n ≤ mY) : |
| Tendsto (fun n => conditionalSigmaEntropy (mY := mY) μ P (m n)) atTop |
| (𝓝 (conditionalSigmaEntropy (mY := mY) μ P (⨆ n, m n))) := by |
| let ℱ : Filtration ℕ mY := ⟨m,hm,hle⟩ |
| let g (a : α) : Y → ℝ := {z | P z=a}.indicator (fun _ => 1) |
| let F : ℕ → Y → ℝ := fun n y => finiteShannon (fun a => (μ[g a | m n]) y) |
| let G : Y → ℝ := fun y => finiteShannon (fun a => (μ[g a | ⨆ n, m n]) y) |
| have hFi n : Integrable (F n) μ := by |
| change Integrable (fun y => ∑ a, Real.negMulLog ((μ[g a | m n]) y)) μ |
| apply integrable_finsetSum |
| intro a _ |
| exact integrable_negMulLog_of_unit_bounds (mY := mY) μ |
| (stronglyMeasurable_condExp.mono (hle n)).aestronglyMeasurable |
| (conditionalProbability_unit_bounds (mY := mY) μ {z | P z=a} (m n)) |
| have hbound n : ∀ᵐ y ∂μ, ‖F n y‖ ≤ (Fintype.card α:ℝ) := by |
| filter_upwards [ae_all_iff.mpr (fun a => |
| conditionalProbability_unit_bounds (mY := mY) μ {z | P z=a} (m n))] with y hy |
| exact finiteShannon_norm_le_card_of_unit_bounds _ hy |
| have hcoord (a : α) : ∀ᵐ y ∂μ, Tendsto (fun n => (μ[g a | m n]) y) atTop |
| (𝓝 ((μ[g a | ⨆ n, m n]) y)) := by |
| exact MeasureTheory.tendsto_ae_condExp (μ := μ) (ℱ := ℱ) (g a) |
| have hlim : ∀ᵐ y ∂μ, Tendsto (fun n => F n y) atTop (𝓝 (G y)) := by |
| filter_upwards [ae_all_iff.mpr hcoord] with y hy |
| have H := continuous_finiteShannon.continuousAt.tendsto.comp (tendsto_pi_nhds.mpr hy) |
| simpa only [F,G,Function.comp_def] using H |
| have H := tendsto_integral_of_dominated_convergence |
| (fun _ : Y => (Fintype.card α:ℝ)) (fun n => (hFi n).aestronglyMeasurable) |
| (integrable_const _) hbound hlim |
| simpa only [conditionalSigmaEntropy,F,G,g] using H |
|
|
| end DukePrimeDegree |
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| end |
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| end OAI |
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|
