import OAI.Probability.FreeIsing.Martingales.ContinuousMaximal

namespace OAI

noncomputable section
open MeasureTheory Filter Set
open scoped NNReal ENNReal Topology

namespace Problem367.MartingaleSquareMaximal

variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω}
  {M : ℕ → ℝ≥0 → Ω → ℝ}

theorem ae_exists_continuous_limit_of_summable_crossings
    (hc : ∀ n, ∀ᵐ ω ∂μ, Continuous (fun t => M n t ω)) (T : ℝ≥0)
    (hsum : (∑' n : ℕ, μ {ω | ∃ t ≤ T,
      ((1 / 2 : ℝ) ^ n) < |M (n + 1) t ω - M n t ω|}) ≠ ∞) :
    ∀ᵐ ω ∂μ, ∃ f : C(Set.Iic T, ℝ),
      ∀ t : Set.Iic T, Tendsto (fun n => M n t ω) atTop (𝓝 (f t)) := by
  have hbc := ae_eventually_notMem hsum
  filter_upwards [ae_all_iff.mpr hc, hbc] with ω hcω hω
  have hcompact : IsCompact (Set.Iic T) := by
    convert (isCompact_Icc (a := (0 : ℝ≥0)) (b := T)) using 1
    ext t
    simp only [Set.mem_Iic, Set.mem_Icc]
    exact ⟨fun ht => ⟨zero_le, ht⟩, And.right⟩
  let : CompactSpace (Set.Iic T) := isCompact_iff_compactSpace.mp hcompact
  let f : ℕ → C(Set.Iic T, ℝ) := fun n =>
    ⟨fun t => M n t ω, (hcω n).comp continuous_subtype_val⟩
  have hgeom : Summable (fun n : ℕ => (1 / 2 : ℝ) ^ n) :=
    summable_geometric_of_norm_lt_one (by norm_num)
  have hdist : Summable (fun n : ℕ => dist (f n) (f (n + 1))) := by
    apply hgeom.of_norm_bounded_eventually_nat
    filter_upwards [hω] with n hn
    rw [Real.norm_eq_abs, abs_of_nonneg dist_nonneg]
    apply (ContinuousMap.dist_le (by positivity : (0 : ℝ) ≤ (1 / 2 : ℝ) ^ n)).2
    intro t
    have hv : |M (n + 1) t ω - M n t ω| ≤ (1 / 2 : ℝ) ^ n := by
      by_contra! h
      exact hn ⟨t, t.property, h⟩
    simpa only [f, ContinuousMap.coe_mk, Real.dist_eq, abs_sub_comm] using hv
  obtain ⟨g, hg⟩ := cauchySeq_tendsto_of_complete (cauchySeq_of_summable_dist hdist)
  refine ⟨g, fun t => ?_⟩
  exact ((continuous_eval_const t).tendsto g).comp hg

theorem ae_exists_continuous_limit_of_summable_energy [IsFiniteMeasure μ]
    {F : Filtration ℝ≥0 mΩ}
    (hM : ∀ n, Martingale (M n) F μ)
    (h2 : ∀ n t, MemLp (M n t) 2 μ)
    (hc : ∀ n, ∀ᵐ ω ∂μ, Continuous (fun t => M n t ω)) (T : ℝ≥0)
    (hsum : (∑' n : ℕ,
      ENNReal.ofReal (∫ ω, (M (n + 1) T ω - M n T ω) ^ 2 ∂μ) /
        (((1 / 2 : ℝ≥0) ^ n : ℝ≥0) : ℝ≥0∞) ^ 2) ≠ ∞) :
    ∀ᵐ ω ∂μ, ∃ f : C(Set.Iic T, ℝ),
      ∀ t : Set.Iic T, Tendsto (fun n => M n t ω) atTop (𝓝 (f t)) := by
  apply ae_exists_continuous_limit_of_summable_crossings hc T
  apply ne_top_of_le_ne_top hsum
  apply ENNReal.tsum_le_tsum
  intro n
  have hcont : ∀ᵐ ω ∂μ, Continuous
      (fun t => (M (n + 1) - M n) t ω) := by
    filter_upwards [hc (n + 1), hc n] with ω ha hb
    exact ha.sub hb
  have hb := continuous_maximal_sq ((hM (n + 1)).sub (hM n))
    (fun t => (h2 (n + 1) t).sub (h2 n t)) hcont ((1 / 2 : ℝ≥0) ^ n) T
  apply (ENNReal.le_div_iff_mul_le (Or.inl (by positivity))
    (Or.inl (by simp))).2
  simpa only [NNReal.coe_pow, NNReal.coe_div, NNReal.coe_one, NNReal.coe_ofNat,
    Pi.sub_apply, mul_comm] using hb

end Problem367.MartingaleSquareMaximal

end

end OAI