| import OAI.Analysis.FiniteFactor.Coupling.Prefix |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Topology ENNReal |
| noncomputable section |
| namespace CouplingParameters |
|
|
| /-- Uniformly square-integrable finite endpoint densities produce an actual |
| infinite-product density, with the same square-integral bound. -/ |
| theorem infiniteEndpoint_density_of_prefix_bound {ι : Type*} |
| (p : ι → CouplingParameters) (right : ι → Bool) (I : ℕ → Finset ι) |
| (hmono : Monotone I) (hcover : ∀ J : Finset ι, ∃ n, J ⊆ I n) |
| (C : ℝ) (hb : ∀ n, (∫ x, prefixDensity p right (I n) x ^ 2 |
| ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) ≤ C) : |
| ∃ g : (ι → ℝ) → ℝ, |
| MemLp g 2 (Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) ∧ |
| (∀ᵐ x ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure), 0 ≤ g x) ∧ |
| (∫ x, g x ^ 2 ∂Measure.infinitePi (fun _ : ι => unitIntervalMeasure)) ≤ C ∧ |
| Measure.infinitePi (fun i => (p i).selectedLaw true (right i)) = |
| (Measure.infinitePi (fun _ : ι => unitIntervalMeasure)).withDensity |
| (fun x => ENNReal.ofReal (g x)) := by |
| let μ := Measure.infinitePi (fun _ : ι => unitIntervalMeasure) |
| obtain ⟨g, hg, hnon, hgb, hsets⟩ := exists_nonneg_L2_limit |
| (fun n => prefixDensity p right (I n)) |
| (fun n => prefixDensity_memLp p right (I n)) |
| (fun n => Eventually.of_forall (prefixDensity_nonneg p right (I n))) C hb |
| (fun n m hnm => prefixDensity_inner p right (hmono hnm)) |
| refine ⟨g, hg, hnon, hgb, ?_⟩ |
| apply (Measure.isProjectiveLimit_infinitePi (fun i => (p i).selectedLaw true (right i))).unique |
| intro J |
| apply Measure.ext |
| intro S hS |
| rw [Measure.map_apply (measurable_restrict J) hS, |
| withDensity_apply _ (hS.preimage (measurable_restrict J)), |
| ← ofReal_integral_eq_lintegral_ofReal (hg.integrable (by norm_num)).restrict |
| (ae_restrict_of_ae hnon)] |
| obtain ⟨N, hN⟩ := hcover J |
| have he : ∀ n ≥ N, ENNReal.ofReal (∫ x in (J.restrict (π := fun _ : ι => ℝ)) ⁻¹' S, |
| prefixDensity p right (I n) x ∂μ) = |
| Measure.pi (fun i : J => (p i).selectedLaw true (right i)) S := by |
| intro n hn |
| exact prefixDensity_cylinder_of_subset p right (hN.trans (hmono hn)) hS |
| have ht := (ENNReal.continuous_ofReal.tendsto _).comp |
| (hsets ((J.restrict (π := fun _ : ι => ℝ)) ⁻¹' S) (hS.preimage (measurable_restrict J))) |
| exact tendsto_nhds_unique ht (tendsto_atTop_of_eventually_const he) |
|
|
| end CouplingParameters |
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
