openAI-math-dataset / lean /OAI /NumberTheory /DukePrimeDegree /Entropy /BilateralConditionalSupport.lean
| 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 | |