| 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 |
|
|
