| import OAI.Probability.InvariantIsing.Core.Variational |
|
|
| namespace OAI |
|
|
| /-! Sequential L1 compactness of bounded monotone overlap paths. The |
| proof extracts the values on the rationals and discards only the |
| countable discontinuity set of the resulting monotone path. -/ |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped Topology |
|
|
| namespace InvariantIsing |
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|
| private def rationalLowerValues (f : ℚ → Icc (0 : ℝ) 1) (x : ℝ) : Set ℝ := |
| (fun q : ℚ => (f q : ℝ)) '' {q : ℚ | (q : ℝ) < x} |
|
|
| private lemma rationalLowerValues_nonempty (f : ℚ → Icc (0 : ℝ) 1) (x : ℝ) : |
| (rationalLowerValues f x).Nonempty := by |
| obtain ⟨q, hq⟩ := exists_rat_lt x |
| exact ⟨f q, q, hq, rfl⟩ |
|
|
| private lemma rationalLowerValues_bdd (f : ℚ → Icc (0 : ℝ) 1) (x : ℝ) : |
| BddAbove (rationalLowerValues f x) := by |
| refine ⟨1, ?_⟩ |
| rintro _ ⟨q, _, rfl⟩ |
| exact (f q).property.2 |
|
|
| private def rationalLowerPath (f : ℚ → Icc (0 : ℝ) 1) : OverlapPath where |
| val x := sSup (rationalLowerValues f x) |
| monotone := by |
| intro x y hxy |
| apply csSup_le_csSup (rationalLowerValues_bdd f y) (rationalLowerValues_nonempty f x) |
| rintro _ ⟨q, hq, rfl⟩ |
| exact ⟨q, hq.trans_le hxy, rfl⟩ |
| nonneg x := by |
| obtain ⟨q, hq⟩ := exists_rat_lt x |
| exact (f q).property.1.trans (le_csSup (rationalLowerValues_bdd f x) ⟨q, hq, rfl⟩) |
| le_one x := by |
| apply csSup_le (rationalLowerValues_nonempty f x) |
| rintro _ ⟨q, _, rfl⟩ |
| exact (f q).property.2 |
|
|
| private lemma rational_le_lowerPath (f : ℚ → Icc (0 : ℝ) 1) |
| {q : ℚ} {x : ℝ} (hq : (q : ℝ) < x) : (f q : ℝ) ≤ rationalLowerPath f x := |
| le_csSup (rationalLowerValues_bdd f x) ⟨q, hq, rfl⟩ |
|
|
| private lemma rationalLowerPath_tendsto (p : ℕ → OverlapPath) |
| (f : ℚ → Icc (0 : ℝ) 1) |
| (hf : ∀ q : ℚ, Tendsto (fun n => p n (q : ℝ)) atTop (𝓝 (f q : ℝ))) |
| (x : ℝ) (hx : ContinuousAt (rationalLowerPath f) x) : |
| Tendsto (fun n => p n x) atTop (𝓝 (rationalLowerPath f x)) := by |
| apply tendsto_order.2 |
| constructor |
| · intro a ha |
| obtain ⟨y, hy, hay⟩ := (lt_csSup_iff (rationalLowerValues_bdd f x) |
| (rationalLowerValues_nonempty f x)).mp ha |
| obtain ⟨q, hqx, rfl⟩ := hy |
| filter_upwards [(hf q).eventually (Ioi_mem_nhds hay)] with n hn |
| exact hn.trans_le ((p n).monotone hqx.le) |
| · intro b hb |
| have hnear : {y : ℝ | rationalLowerPath f y < b} ∈ 𝓝 x := |
| hx (Iio_mem_nhds hb) |
| obtain ⟨ε, hε, hball⟩ := Metric.mem_nhds_iff.mp hnear |
| let y := x + ε / 2 |
| have hxy : x < y := by dsimp [y]; linarith |
| have hy : rationalLowerPath f y < b := by |
| apply hball |
| rw [Metric.mem_ball, Real.dist_eq, abs_of_pos (sub_pos.mpr hxy)] |
| dsimp [y] |
| linarith |
| obtain ⟨q, hxq, hqy⟩ := exists_rat_btwn hxy |
| have hqb : (f q : ℝ) < b := (rational_le_lowerPath f hqy).trans_lt hy |
| filter_upwards [(hf q).eventually (Iio_mem_nhds hqb)] with n hn |
| exact ((p n).monotone hxq.le).trans_lt hn |
|
|
| / |
| with another admissible bounded monotone representative. -/ |
| theorem overlapPath_ae_l1_subsequence (p : ℕ → OverlapPath) : |
| ∃ q : OverlapPath, ∃ φ : ℕ → ℕ, StrictMono φ ∧ |
| (∀ᵐ s ∂pathMeasure, Tendsto (fun n => p (φ n) s) atTop (𝓝 (q s))) ∧ |
| Tendsto (fun n => ∫ s, |p (φ n) s - q s| ∂pathMeasure) atTop (𝓝 0) := by |
| let f : ℕ → ℚ → Icc (0 : ℝ) 1 := fun n r => ⟨p n r, (p n).nonneg r, (p n).le_one r⟩ |
| obtain ⟨g, φ, hφ, hg⟩ := CompactSpace.tendsto_subseq f |
| let q := rationalLowerPath g |
| have hr (r : ℚ) : Tendsto (fun n => p (φ n) (r : ℝ)) atTop (𝓝 (g r : ℝ)) := |
| (continuous_subtype_val.tendsto (g r)).comp ((continuous_apply r).tendsto g |>.comp hg) |
| have hc : ∀ᵐ s ∂pathMeasure, ContinuousAt q s := by |
| let : NullSingletonClass pathMeasure := by unfold pathMeasure; infer_instance |
| have hz := q.monotone.countable_not_continuousAt.measure_zero (μ := pathMeasure) |
| have hae : ∀ᵐ s ∂pathMeasure, s ∉ {x | ¬ContinuousAt q x} := compl_mem_ae_iff.mpr hz |
| filter_upwards [hae] with s hs |
| exact not_not.mp hs |
| have hpoint : ∀ᵐ s ∂pathMeasure, Tendsto (fun n => p (φ n) s) atTop (𝓝 (q s)) := by |
| filter_upwards [hc] with s hs |
| exact rationalLowerPath_tendsto (fun n => p (φ n)) g hr s hs |
| refine ⟨q, φ, hφ, hpoint, ?_⟩ |
| have ht := tendsto_integral_of_dominated_convergence (μ := pathMeasure) |
| (F := fun n s => |p (φ n) s - q s|) (f := fun _ => (0 : ℝ)) (fun _ => (1 : ℝ)) |
| (fun n => ((p (φ n)).measurable.sub q.measurable).abs.aestronglyMeasurable) |
| (integrable_const _) (fun n => ae_of_all _ fun s => by |
| rw [Real.norm_eq_abs, abs_abs] |
| exact abs_le.mpr ⟨by linarith [(p (φ n)).nonneg s, q.le_one s], |
| by linarith [(p (φ n)).le_one s, q.nonneg s]⟩) (by |
| filter_upwards [hpoint] with s hs |
| simpa only [sub_self, abs_zero] using (hs.sub_const (q s)).abs) |
| simpa only [integral_zero] using ht |
|
|
| theorem overlapPath_l1_subsequence (p : ℕ → OverlapPath) : |
| ∃ q : OverlapPath, ∃ φ : ℕ → ℕ, StrictMono φ ∧ |
| Tendsto (fun n => ∫ s, |p (φ n) s - q s| ∂pathMeasure) atTop (𝓝 0) := by |
| obtain ⟨q, φ, hφ, _, hq⟩ := overlapPath_ae_l1_subsequence p |
| exact ⟨q, φ, hφ, hq⟩ |
|
|
| end InvariantIsing |
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| end |
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| end OAI |
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