import Mathlib.MeasureTheory.OuterMeasure.BorelCantelli
import Mathlib.MeasureTheory.Function.Floor
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.Algebra.InfiniteSum.ENNReal
import Mathlib.Tactic

namespace OAI

open Set MeasureTheory Filter
open scoped ENNReal

namespace Invariant233.CocycleSupport

theorem uniform_floor_ne_zero_le (t : ℝ) :
    (volume.restrict (Ico (0 : ℝ) 1)) {x : ℝ | ⌊x + t⌋ ≠ (0 : ℤ)} ≤
      ENNReal.ofReal |t| := by
  have hm : MeasurableSet {x : ℝ | ⌊x + t⌋ ≠ (0 : ℤ)} :=
    ((measurable_id.add_const t).floor (measurableSet_singleton 0)).compl
  rw [Measure.restrict_apply hm]
  rcases le_or_gt 0 t with ht | ht
  · calc
      volume ({x : ℝ | ⌊x + t⌋ ≠ (0 : ℤ)} ∩ Ico 0 1) ≤
          volume (Ico (1 - t) 1) := by
        apply measure_mono
        rintro x ⟨hx, hx0, hx1⟩
        refine ⟨?_, hx1⟩
        by_contra h
        apply hx
        apply Int.floor_eq_zero_iff.mpr
        constructor <;> linarith
      _ = ENNReal.ofReal |t| := by
        rw [Real.volume_Ico, abs_of_nonneg ht]
        congr 1
        ring
  · calc
      volume ({x : ℝ | ⌊x + t⌋ ≠ (0 : ℤ)} ∩ Ico 0 1) ≤
          volume (Ico 0 (-t)) := by
        apply measure_mono
        rintro x ⟨hx, hx0, hx1⟩
        refine ⟨hx0, ?_⟩
        by_contra h
        apply hx
        apply Int.floor_eq_zero_iff.mpr
        constructor <;> linarith
      _ = ENNReal.ofReal |t| := by
        rw [Real.volume_Ico, sub_zero, abs_of_neg ht]

theorem ae_finite_support_of_tsum_ne_top {Ω ι A : Type*}
    [MeasurableSpace Ω] [Countable ι] [Zero A]
    (μ : Measure Ω) (a : ι → Ω → A)
    (h : (∑' i, μ {x | a i x ≠ 0}) ≠ ∞) :
    ∀ᵐ x ∂μ, (Function.support (fun i => a i x)).Finite := by
  exact ae_finite_setOfPred_mem h

theorem ae_all_finite_support_of_tsum_ne_top {Ω ι K A : Type*}
    [MeasurableSpace Ω] [Countable ι] [Countable K] [Zero A]
    (μ : Measure Ω) (a : K → ι → Ω → A)
    (h : ∀ k, (∑' i, μ {x | a k i x ≠ 0}) ≠ ∞) :
    ∀ᵐ x ∂μ, ∀ k, (Function.support (fun i => a k i x)).Finite := by
  rw [ae_all_iff]
  intro k
  exact ae_finite_support_of_tsum_ne_top μ (a k) (h k)

theorem measure_floor_ne_zero_le {Ω : Type*} [MeasurableSpace Ω]
    (μ : Measure Ω) (u : Ω → ℝ) (hu : Measurable u)
    (hlaw : μ.map u = volume.restrict (Ico (0 : ℝ) 1)) (t : ℝ) :
    μ {x | ⌊u x + t⌋ ≠ (0 : ℤ)} ≤ ENNReal.ofReal |t| := by
  have hm : MeasurableSet {x : ℝ | ⌊x + t⌋ ≠ (0 : ℤ)} :=
    ((measurable_id.add_const t).floor (measurableSet_singleton 0)).compl
  calc
    μ {x | ⌊u x + t⌋ ≠ (0 : ℤ)} =
        (μ.map u) {x : ℝ | ⌊x + t⌋ ≠ (0 : ℤ)} :=
      (Measure.map_apply hu hm).symm
    _ = (volume.restrict (Ico (0 : ℝ) 1))
        {x : ℝ | ⌊x + t⌋ ≠ (0 : ℤ)} := by rw [hlaw]
    _ ≤ ENNReal.ofReal |t| := uniform_floor_ne_zero_le t

theorem ae_all_int_finite_floor_support {Ω ι : Type*}
    [MeasurableSpace Ω] [Countable ι]
    (μ : Measure Ω) (u : ι → Ω → ℝ) (hu : ∀ i, Measurable (u i))
    (hlaw : ∀ i, μ.map (u i) = volume.restrict (Ico (0 : ℝ) 1))
    (θ : ι → ℝ) (hθ0 : ∀ i, 0 ≤ θ i) (hθ : Summable θ) :
    ∀ᵐ x ∂μ, ∀ k : ℤ,
      (Function.support (fun i => (⌊u i x + (k : ℝ) * θ i⌋ : ℤ))).Finite := by
  apply ae_all_finite_support_of_tsum_ne_top
  intro k
  apply ne_top_of_le_ne_top ((hθ.mul_left |(k : ℝ)|).tsum_ofReal_ne_top)
  apply ENNReal.tsum_le_tsum
  intro i
  simpa only [abs_mul, abs_of_nonneg (hθ0 i)] using
    measure_floor_ne_zero_le μ (u i) (hu i) (hlaw i) ((k : ℝ) * θ i)

end Invariant233.CocycleSupport

end OAI