| import Mathlib |
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| namespace OAI |
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| section |
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| open Set Filter Metric |
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| namespace CAT0Fillings.JointBV |
| universe u v |
| lemma totallyBounded_range_of_finite_coordinate_control {ι : Type*} {α : Type u} |
| [PseudoMetricSpace α] {β : ℕ → Type v} [∀ i, PseudoMetricSpace (β i)] |
| [∀ i, CompleteSpace (β i)] {f : ι → α} {g : ∀ i, ι → β i} |
| (hg : ∀ i, TotallyBounded (range (g i))) |
| (hcontrol : ∀ ε > 0, ∃ N : ℕ, ∃ δ > 0, |
| ∀ a b, (∀ i : Fin N, dist (g i a) (g i b) < δ) → dist (f a) (f b) < ε) : |
| TotallyBounded (range f) := by |
| classical |
| apply Metric.totallyBounded_of_finite_discretization |
| intro ε hε |
| obtain ⟨N,δ,hδ,hc⟩ := hcontrol ε hε |
| let G (a : ι) (i : Fin N) := g i a |
| have hG : TotallyBounded (range G) := by |
| apply (isCompact_pi_infinite (fun i : Fin N => ((hg i).closure.isCompact_of_isClosed isClosed_closure))).totallyBounded.subset |
| rintro _ ⟨a,rfl⟩ i |
| exact subset_closure (mem_range_self a) |
| obtain ⟨s,hs,hfin,hcover⟩ := Metric.finite_approx_of_totallyBounded hG (δ/2) (half_pos hδ) |
| have : Fintype s := hfin.fintype |
| have hnear (a : ι) : ∃ y : s, dist (G a) y < δ/2 := by |
| have := hcover (mem_range_self a) |
| simp only [mem_iUnion,mem_ball] at this |
| obtain ⟨y,hy,hay⟩ := this |
| exact ⟨⟨y,hy⟩,hay⟩ |
| choose c hc' using hnear |
| let pick (x : range f) : ι := x.property.choose |
| have hp (x : range f) : f (pick x) = x := x.property.choose_spec |
| refine ⟨ULift.{u} (Fin (Fintype.card s)),inferInstance,fun x => ULift.up ((Fintype.equivFin s) (c (pick x))),?_⟩ |
| intro x y he |
| have he : c (pick x) = c (pick y) := (Fintype.equivFin s).injective (congrArg ULift.down he) |
| rw [←hp x,←hp y] |
| apply hc |
| intro i |
| apply (dist_le_pi_dist (G (pick x)) (G (pick y)) i).trans_lt |
| calc dist (G (pick x)) (G (pick y)) ≤ |
| dist (G (pick x)) (c (pick x)) + dist (G (pick y)) (c (pick x)) := dist_triangle_right _ _ _ |
| _ < δ/2+δ/2 := add_lt_add (hc' _) (by rw [he]; exact hc' _) |
| _ = δ := add_halves _ |
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| end CAT0Fillings.JointBV |
| end |
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| section |
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| open Set Filter MeasureTheory |
| open scoped Topology ENNReal |
|
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| namespace CAT0Fillings.JointBV |
| variable {α : Type*} [MeasurableSpace α] (μ : Measure α) [IsFiniteMeasure μ] |
| (ρ : α → ℕ) (hρ : Measurable ρ) |
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| def indexTail (N : ℕ) (x : α) : ℝ := if N ≤ ρ x then 1 else 0 |
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| include hρ |
| lemma measurable_indexTail (N : ℕ) : Measurable (indexTail ρ N) := |
| measurable_const.ite (measurableSet_le measurable_const hρ) measurable_const |
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| omit hρ in |
| lemma indexTail_bound {α : Type*} [MeasurableSpace α] (ρ : α → ℕ) |
| (N : ℕ) (x : α) : |indexTail ρ N x| ≤ 1 := by |
| unfold indexTail; split_ifs <;> norm_num |
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| lemma memLp_indexTail (N : ℕ) : MemLp (indexTail ρ N) 2 μ := |
| MemLp.of_bound (measurable_indexTail ρ hρ N).aestronglyMeasurable 1 |
| (Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using indexTail_bound ρ N x) |
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| omit hρ in |
| lemma lpNorm_indexTail {α : Type*} [MeasurableSpace α] (μ : Measure α) |
| [IsFiniteMeasure μ] (ρ : α → ℕ) (hρ : Measurable ρ) (N : ℕ) : |
| lpNorm (indexTail ρ N) 2 μ = (μ.real {x | N ≤ ρ x}) ^ (1/(2:ℝ)) := by |
| rw [←toReal_eLpNorm] |
| have he : indexTail ρ N = {x | N ≤ ρ x}.indicator (fun _ => (1:ℝ)) := by |
| ext x; simp only [indexTail,Set.indicator_apply,mem_ofPred_eq] |
| rw [he, eLpNorm_indicator_const (s := {x | N ≤ ρ x}) |
| (measurableSet_le measurable_const hρ).nullMeasurableSet (by norm_num) (by norm_num)] |
| simp [ENNReal.toReal_rpow,measureReal_def] |
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| lemma tendsto_lpNorm_indexTail : Tendsto (fun N => lpNorm (indexTail ρ N) 2 μ) atTop (𝓝 0) := by |
| have hsets : (⋂ N : ℕ, {x | N ≤ ρ x}) = ∅ := by |
| ext x |
| simp only [mem_iInter,mem_ofPred_eq,mem_empty_iff_false,iff_false] |
| intro hx |
| exact Nat.not_succ_le_self (ρ x) (hx (ρ x+1)) |
| have hm : Tendsto (fun N => μ {x | N ≤ ρ x}) atTop (𝓝 0) := by |
| simpa only [Function.comp_def,hsets,measure_empty] using |
| tendsto_measure_iInter_atTop (μ := μ) (fun N => (measurableSet_le measurable_const hρ).nullMeasurableSet) |
| (show Antitone (fun N : ℕ => {x | N ≤ ρ x}) by intro N M hNM x hx; exact hNM.trans hx) |
| ⟨0,measure_ne_top _ _⟩ |
| have hr : Tendsto (fun N => μ.real {x | N ≤ ρ x}) atTop (𝓝 0) := by |
| simpa only [measureReal_def,ENNReal.toReal_zero,Function.comp_def] using (ENNReal.continuousAt_toReal (by simp : (0:ℝ≥0∞) ≠ ∞)).tendsto.comp hm |
| simpa only [lpNorm_indexTail μ ρ hρ,Real.zero_rpow (by norm_num : (1/(2:ℝ)) ≠ 0)] using |
| (hr.rpow_const (by norm_num : (0:ℝ) ≠ 0 ∨ 0 ≤ 1/(2:ℝ))) |
|
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| end CAT0Fillings.JointBV |
| end |
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| end OAI |
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