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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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | 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
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