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namespace OAI
/-! # Chain rules with one countable conditioning label
Finite clipping of the conditioning label and the martingale limit extend
the finite chain rule without requiring finite entropy of that label.
-/
noncomputable section
open MeasureTheory Set Filter
open scoped Topology
namespace DukePrimeDegree
open PrimeDegreePackets
lemma observationSigma_pair_eq {Y α β : Type*} [Countable α] [Countable β]
(P : Y → α) (Q : Y → β) :
observationSigma (fun y => (P y,Q y))=observationSigma P ⊔ observationSigma Q := by
let : MeasurableSpace α := ⊤
let : MeasurableSpace β := ⊤
have hprod : (inferInstance : MeasurableSpace (α × β))=⊤ := by
apply top_unique
intro s _
exact (Set.to_countable s).measurableSet
unfold observationSigma
rw [←hprod]
exact MeasurableSpace.comap_prodMk P Q
theorem conditionalSigmaEntropy_chain_nat {Y α β : Type*}
[MeasurableSpace Y] [Fintype α] [Fintype β]
(μ : Measure Y) [IsProbabilityMeasure μ]
(P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a})
(R : Y → β) (hR : ∀ b, MeasurableSet {y | R y=b})
(F : Y → ℕ) (hF : Measurable F) :
conditionalSigmaEntropy μ (fun y => (P y,R y))
(MeasurableSpace.comap F inferInstance) =
conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) +
conditionalSigmaEntropy μ R
(MeasurableSpace.comap F inferInstance ⊔ observationSigma P) := by
let C := fun L : ℕ => fun y : Y => (heightClip F L y,P y)
have hC (L : ℕ) (b : Fin (L+1) × α) : MeasurableSet {y | C L y=b} :=
observationPair_measurable_fiber _ _
(fun a => heightClip_measurable hF L (measurableSet_singleton a)) hP b
have hCm : Monotone (fun L => observationSigma (C L)) := by
intro L J hLJ
simp only [C,observationSigma_pair_eq]
exact sup_le_sup_right (heightClip_sigma_monotone F hLJ) _
have hCg : (⨆ L, observationSigma (C L))=
MeasurableSpace.comap F inferInstance ⊔ observationSigma P := by
simp only [C,observationSigma_pair_eq]
rw [iSup_sup_eq,heightClip_sigma_generate,iSup_const]
have h₁ := conditional_heightClip_tendsto μ (fun y => (P y,R y))
(observationPair_measurable_fiber P R hP hR) F hF
have h₂ := conditional_heightClip_tendsto μ P hP F hF
have h₃ := conditionalObservationEntropy_tendsto_sigma μ R hR C hC hCm
rw [hCg] at h₃
apply tendsto_nhds_unique h₁
apply (h₂.add h₃).congr
intro L
exact (conditionalObservationEntropy_chain μ P R (heightClip F L)).symm
theorem conditionalSigmaEntropy_comp_preserving_nat {Y α : Type*}
[MeasurableSpace Y] [Fintype α]
(μ : Measure Y) [IsProbabilityMeasure μ]
{T : Y → Y} (hT : MeasurePreserving T μ μ)
(P : Y → α) (hP : ∀ a, MeasurableSet {y | P y=a})
(F : Y → ℕ) (hF : Measurable F) :
conditionalSigmaEntropy μ (P ∘ T) (MeasurableSpace.comap (F ∘ T) inferInstance) =
conditionalSigmaEntropy μ P (MeasurableSpace.comap F inferInstance) := by
have h₁ := conditional_heightClip_tendsto μ (P ∘ T)
(fun a => (hP a).preimage hT.measurable) (F ∘ T) (hF.comp hT.measurable)
have h₂ := conditional_heightClip_tendsto μ P hP F hF
apply tendsto_nhds_unique h₁
apply h₂.congr
intro L
symm
exact conditionalObservationEntropy_comp_preserving μ P (heightClip F L) hP
(fun a => heightClip_measurable hF L (measurableSet_singleton a)) hT
end DukePrimeDegree
end
end OAI
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