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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 | import OAI.NumberTheory.DukePrimeDegree.Entropy.NeutralFiberNames
namespace OAI
/-! # Actual conditional measures on bilateral itinerary fibers -/
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators
namespace DukePrimeDegree
open PrimeDegreePackets
/-- Disintegration of the graph of a measurable map is concentrated on
the corresponding fibers. -/
theorem ae_graph_condKernel_fiber
{S Y : Type*} [MeasurableSpace S] [MeasurableEq S]
[MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y]
(μ : Measure Y) [IsProbabilityMeasure μ]
(f : Y → S) (hf : Measurable f) :
let ρ := μ.map (fun y => (f y,y))
∀ᵐ s ∂ρ.fst, ∀ᵐ y ∂ρ.condKernel s, f y=s := by
let ρ := μ.map (fun y => (f y,y))
have hg : ∀ᵐ z ∂ρ, f z.2=z.1 := by
apply (ae_map_iff (hf.prodMk measurable_id).aemeasurable
(measurableSet_eq_fun (hf.comp measurable_snd) measurable_fst)).mpr
exact Filter.Eventually.of_forall fun _ => rfl
have hcomp : ∀ᵐ z ∂ρ.fst ⊗ₘ ρ.condKernel, f z.2=z.1 := by
rwa [Measure.disintegrate]
exact Measure.ae_ae_of_ae_compProd hcomp
theorem ae_condDistrib_self_fiber
{S Y : Type*} [MeasurableSpace S] [MeasurableEq S]
[MeasurableSpace Y] [StandardBorelSpace Y] [Nonempty Y]
(μ : Measure Y) [IsProbabilityMeasure μ]
(f : Y → S) (hf : Measurable f) :
∀ᵐ s ∂μ.map f, ∀ᵐ y ∂condDistrib id f μ s, f y=s := by
have H := ae_graph_condKernel_fiber μ f hf
change ∀ᵐ s ∂(μ.map (fun y => (f y,y))).fst,
∀ᵐ y ∂(μ.map (fun y => (f y,y))).condKernel s, f y=s at H
have hm : (μ.map (fun y => (f y,y))).fst=μ.map f :=
Measure.fst_map_prodMk hf measurable_id
rw [hm] at H
simpa only [condDistrib,id_eq] using H
def bilateralNameObservation {n : ℕ}
(ell : Fin n → ℝ) (hell : ∑ i, ell i=0) (Q : X n → ℕ)
(x : X n) : (ℕ → ℕ) × (ℕ → ℕ) :=
(fun k => Q ((fun z => rightAction z (diagonalOneParameter ell hell (-1)))^[k] x),
fun k => Q ((fun z => rightAction z (diagonalOneParameter ell hell 1))^[k] x))
theorem measurable_bilateralNameObservation {n : ℕ}
(ell : Fin n → ℝ) (hell : ∑ i, ell i=0)
(Q : X n → ℕ) (hQ : Measurable Q) :
Measurable (bilateralNameObservation ell hell Q) := by
have hT (t : ℝ) : Measurable (fun z : X n =>
rightAction z (diagonalOneParameter ell hell t)) :=
(continuous_rightAction _).measurable
exact (measurable_pi_iff.mpr fun k => hQ.comp ((hT (-1)).iterate k)).prodMk
(measurable_pi_iff.mpr fun k => hQ.comp ((hT 1).iterate k))
theorem bilateralNameObservation_eq_iff {n : ℕ}
(ell : Fin n → ℝ) (hell : ∑ i, ell i=0)
(Q : X n → ℕ) (x y : X n) :
bilateralNameObservation ell hell Q y = bilateralNameObservation ell hell Q x ↔
y ∈ bilateralNameFiber ell hell Q x := by
simp only [bilateralNameObservation,Prod.mk.injEq,funext_iff,bilateralNameFiber,
mem_ofPred_eq]
/-- Almost every conditional measure for the actual bilateral itinerary
law is supported on the geometric fiber of any representative. -/
theorem ae_bilateral_condKernel_fiber {n : ℕ}
(ell : Fin n → ℝ) (hell : ∑ i, ell i=0)
(Q : X n → ℕ) (hQ : Measurable Q)
(μ : Measure (X n)) [IsProbabilityMeasure μ] :
let F := bilateralNameObservation ell hell Q
let ρ := μ.map (fun x => (F x,x))
∀ᵐ s ∂ρ.fst, ∀ x, F x=s →
∀ᵐ y ∂ρ.condKernel s, y ∈ bilateralNameFiber ell hell Q x := by
have h := ae_graph_condKernel_fiber μ _ (measurable_bilateralNameObservation ell hell Q hQ)
filter_upwards [h] with s hs
intro x hx
filter_upwards [hs] with y hy
exact (bilateralNameObservation_eq_iff ell hell Q x y).mp (hy.trans hx.symm)
end DukePrimeDegree
end
end OAI
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