| import OAI.NumberTheory.DukePrimeDegree.Entropy.CountableFutureSigma |
| import Mathlib.Probability.Kernel.MeasurableLIntegral |
|
|
| namespace OAI |
|
|
| /-! # The entire future as a measurable sequence |
|
|
| This representation makes future-measurable events constant on each |
| complete future fiber. It is used to restrict local disintegrations to |
| saturated events when gluing them. |
| -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
|
|
| namespace DukePrimeDegree |
| open PrimeDegreePackets |
|
|
| def entireFutureObservation {Y : Type*} (T : Y → Y) (Q : Y → ℕ) |
| (y : Y) : ℕ → ℕ := fun k => Q (T^[k+1] y) |
|
|
| lemma entireFutureObservation_measurable {Y : Type*} [MeasurableSpace Y] |
| {T : Y → Y} (hT : Measurable T) {Q : Y → ℕ} (hQ : Measurable Q) : |
| Measurable (entireFutureObservation T Q) := |
| measurable_pi_iff.mpr fun k => hQ.comp (hT.iterate (k+1)) |
|
|
| lemma countableFutureSigma_eq_comap {Y : Type*} (T : Y → Y) (Q : Y → ℕ) : |
| (⨆ k : ℕ, observationSigma (fun y => Q (T^[k+1] y))) = |
| MeasurableSpace.comap (entireFutureObservation T Q) |
| (inferInstance : MeasurableSpace (ℕ → ℕ)) := by |
| exact (MeasurableSpace.comap_process_pi |
| (fun k : ℕ => fun y : Y => Q (T^[k+1] y))).symm |
|
|
| lemma transverseCountableFutureSigma_eq_comap {S V Y : Type*} |
| [MeasurableSpace S] [MeasurableSpace V] |
| (T : Y → Y) (Q : Y → ℕ) (F : S × V → Y) : |
| (⨆ m, transverseLabelSigma (fun z => countableFutureCode T Q m (F z))) = |
| MeasurableSpace.comap (fun z => (z.1, entireFutureObservation T Q (F z))) |
| (inferInstance : MeasurableSpace (S × (ℕ → ℕ))) := by |
| rw [transverseCountableFutureSigma_eq_sup] |
| calc |
| _ = MeasurableSpace.comap Prod.fst (inferInstance : MeasurableSpace S) ⊔ |
| MeasurableSpace.comap (entireFutureObservation T Q ∘ F) |
| (inferInstance : MeasurableSpace (ℕ → ℕ)) := by |
| rw [countableFutureSigma_eq_comap, MeasurableSpace.comap_comp] |
| _ = _ := (MeasurableSpace.comap_prodMk |
| (Prod.fst : S × V → S) (entireFutureObservation T Q ∘ F)).symm |
|
|
| lemma countableFutureFiber_eq_of_mem {S V Y : Type*} |
| (T : Y → Y) (Q : Y → ℕ) (F : S × V → Y) |
| (z : S × V) {u : V} (hu : u ∈ countableFutureFiber T Q F z) : |
| countableFutureFiber T Q F (z.1,u) = countableFutureFiber T Q F z := by |
| ext v |
| constructor |
| · intro hv k |
| exact (hv k).trans (hu k) |
| · intro hv k |
| exact (hv k).trans (hu k).symm |
|
|
| /-- Every event in the complete transverse-future sigma algebra is |
| constant along its future fibers, with no exceptional set needed. -/ |
| theorem mem_future_event_iff {S V Y : Type*} |
| [MeasurableSpace S] [MeasurableSpace V] |
| (T : Y → Y) (Q : Y → ℕ) (F : S × V → Y) |
| {E : Set (S × V)} |
| (hE : MeasurableSet[⨆ m, transverseLabelSigma |
| (fun z => countableFutureCode T Q m (F z))] E) |
| (z : S × V) {u : V} (hu : u ∈ countableFutureFiber T Q F z) : |
| (z.1,u) ∈ E ↔ z ∈ E := by |
| rw [transverseCountableFutureSigma_eq_comap] at hE |
| obtain ⟨A, _hA, rfl⟩ := hE |
| change (z.1, entireFutureObservation T Q (F (z.1,u))) ∈ A ↔ |
| (z.1, entireFutureObservation T Q (F z)) ∈ A |
| have he : entireFutureObservation T Q (F (z.1,u)) = |
| entireFutureObservation T Q (F z) := funext hu |
| rw [he] |
|
|
| /-- Fiber masses of a kernel depending only on the transverse coordinate |
| are measurable in the transverse-future sigma algebra itself. Thus mass |
| conditions used to select boxes really are future events. -/ |
| theorem measurable_countableFuture_mass {S V Y : Type*} |
| [MeasurableSpace S] [MeasurableSpace V] [MeasurableSpace Y] |
| (κ : Kernel S V) [IsSFiniteKernel κ] |
| {T : Y → Y} (hT : Measurable T) {Q : Y → ℕ} (hQ : Measurable Q) |
| {F : S × V → Y} (hF : Measurable F) |
| (A : Set (S × V)) (hA : MeasurableSet A) : |
| Measurable[⨆ m, transverseLabelSigma (fun z => countableFutureCode T Q m (F z))] |
| (fun z => κ z.1 ({t | (z.1,t) ∈ A} ∩ countableFutureFiber T Q F z)) := by |
| let q := entireFutureObservation T Q |
| have hq : Measurable q := entireFutureObservation_measurable hT hQ |
| have hrel : MeasurableSet {p : (S × (ℕ → ℕ)) × V | |
| (p.1.1,p.2) ∈ A ∧ q (F (p.1.1,p.2)) = p.1.2} := |
| (hA.preimage (measurable_fst.fst.prodMk measurable_snd)).inter |
| (measurableSet_eq_fun |
| (hq.comp (hF.comp (measurable_fst.fst.prodMk measurable_snd))) measurable_fst.snd) |
| have H : Measurable (fun b : S × (ℕ → ℕ) => |
| κ b.1 {t | (b.1,t) ∈ A ∧ q (F (b.1,t)) = b.2}) := |
| Kernel.measurable_kernel_prodMk_left (κ := κ.prodMkRight (ℕ → ℕ)) hrel |
| have hη : Measurable[⨆ m, transverseLabelSigma |
| (fun z => countableFutureCode T Q m (F z))] |
| (fun z => (z.1,q (F z))) := by |
| rw [transverseCountableFutureSigma_eq_comap] |
| exact comap_measurable _ |
| convert H.comp hη using 1 |
| funext z |
| congr 1 |
| ext t |
| simp only [Set.mem_inter_iff, Set.mem_ofPred_eq, countableFutureFiber, |
| q, entireFutureObservation, funext_iff] |
|
|
| /-- The event that a complete future fiber has no mass outside a fixed |
| time domain is measurable using only the transverse coordinate and the |
| future name. -/ |
| theorem measurableSet_future_mass_contained {S V Y : Type*} |
| [MeasurableSpace S] [MeasurableSpace V] [MeasurableSpace Y] |
| (κ : Kernel S V) [IsSFiniteKernel κ] |
| {T : Y → Y} (hT : Measurable T) {Q : Y → ℕ} (hQ : Measurable Q) |
| {F : S × V → Y} (hF : Measurable F) |
| (B : Set V) (hB : MeasurableSet B) : |
| MeasurableSet[⨆ m, transverseLabelSigma (fun z => countableFutureCode T Q m (F z))] |
| {z | κ z.1 (countableFutureFiber T Q F z \ B) = 0} := by |
| have hm := measurable_countableFuture_mass κ hT hQ hF |
| (Prod.snd ⁻¹' Bᶜ) (hB.compl.preimage measurable_snd) |
| have he := measurableSet_eq_fun hm (measurable_const (a := (0 : ℝ≥0∞))) |
| change MeasurableSet[⨆ m, transverseLabelSigma (fun z => countableFutureCode T Q m (F z))] |
| {z | κ z.1 (Bᶜ ∩ countableFutureFiber T Q F z) = 0} at he |
| simpa only [Set.sdiff_eq, Set.inter_comm] using he |
| |
| end DukePrimeDegree |
| |
| end |
| |
| end OAI |
| |
