File size: 3,036 Bytes
a44267b | 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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | import OAI.NumberTheory.DukePrimeDegree.Entropy.ConditionalSigmaOrder
namespace OAI
/-! # Entropy along increasing conditioning sigma algebras
The martingale limit applies also when the conditioning includes an
arbitrary transverse coordinate in addition to finite future names.
-/
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
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
end
end OAI
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