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ddc84c5 | 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 | 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
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