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