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8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 | import Mathlib
import OAI.Geometry.CAT0Fillings.Slicing.ProfileIntegral
import OAI.Geometry.CAT0Fillings.Mass.Duality
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology NNReal ENNReal BigOperators
namespace CAT0Fillings
namespace BorelCoefficients
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X]
abbrev CoefficientList (N : ℕ) :=
{b : Fin N → C(X,ℝ) // (∀ i, BoundedLip (b i)) ∧ ∀ x, ∑ i, |b i x| ≤ 1}
instance coefficientList_nonempty (N : ℕ) : Nonempty (CoefficientList (X := X) N) := by
refine ⟨⟨fun _ => 0,?_,?_⟩⟩
· intro i
exact ⟨⟨0,LipschitzWith.const 0⟩,0,by simp⟩
· simp
lemma coefficientList_eval_continuous {k N : ℕ} {T : Functional X k} (hT : IsMetricCurrent T)
(π : Fin N → Fin k → X → ℝ) (hπ : ∀ j i, LipschitzWith 1 (π j i)) :
Continuous (fun b : CoefficientList (X := X) N => ∑ i, T (b.val i) (π i)) := by
apply continuous_finsetSum
intro i hi
apply LipschitzWith.continuous (K := ⟨mass T,mass_nonneg _⟩)
apply LipschitzWith.of_dist_le_mul
intro b c
have hh := Foundations.IsMetricCurrent.first_difference_bound hT (b.property.1 i) (c.property.1 i)
(π i) 1 (hπ i) (nndist b c) (fun x =>
(ContinuousMap.dist_apply_le_dist x).trans (dist_le_pi_dist b.val c.val i))
change |T (b.val i) (π i)-T (c.val i) (π i)| ≤ mass T*dist b c
simpa only [Real.dist_eq,NNReal.coe_one,one_pow,one_mul,coe_nndist,mul_comm] using hh
lemma finite_list_le_mass {k N : ℕ} {T : Functional X k} (hT : IsMetricCurrent T)
(b : Fin N → X → ℝ) (hb : ∀ i, BoundedLip (b i))
(hbudget : ∀ x, ∑ i, |b i x| ≤ 1)
(π : Fin N → Fin k → X → ℝ) (hπ : ∀ j i, LipschitzWith 1 (π j i)) :
(∑ i, T (b i) (π i)) ≤ mass T := by
let μ := MassMeasure.currentMassMeasure hT
have hi i : Integrable (fun x => |b i x|) μ := (integrable_boundedLip μ (hb i)).abs
calc (∑ i, T (b i) (π i)) ≤ ∑ i, |T (b i) (π i)| :=
Finset.sum_le_sum fun i _ => le_abs_self _
_ ≤ ∑ i, ∫ x, |b i x| ∂μ := Finset.sum_le_sum fun i _ =>
MassMeasure.currentMassMeasure_controls hT _ _ (hb i) (hπ i)
_ = ∫ x, ∑ i, |b i x| ∂μ := (integral_finsetSum _ (fun i _ => hi i)).symm
_ ≤ ∫ x, (1:ℝ) ∂μ := integral_mono (integrable_finsetSum _ fun i _ => hi i)
(integrable_const _) hbudget
_ = mass T := by simpa using MassMeasure.currentMassMeasure_total hT
theorem exists_countable_mass_lists (k : ℕ) :
∃ (π : ℕ → Fin k → X → ℝ) (b : (N : ℕ) → ℕ → Fin (N+1) → X → ℝ),
(∀ j i, LipschitzWith 1 (π j i)) ∧
(∀ N j i, BoundedLip (b N j i)) ∧
(∀ N j x, ∑ i, |b N j i x| ≤ 1) ∧
∀ {T : Functional X k}, IsMetricCurrent T → ∀ C : ℝ,
mass T ≤ C ↔ ∀ N j, (∑ i : Fin (N+1), T (b N j i) (π i)) ≤ C := by
classical
obtain ⟨π,hπ,hspec⟩ := exists_mass_spectrum (X := X) k
choose bs hbs using fun N : ℕ => TopologicalSpace.exists_dense_seq (CoefficientList (X := X) (N+1))
refine ⟨π,fun N j i => (bs N j).val i,hπ,fun N j => (bs N j).property.1,
fun N j => (bs N j).property.2,?_⟩
intro T hT C
constructor
· intro h N j
exact (finite_list_le_mass hT _ (bs N j).property.1 (bs N j).property.2
(fun i => π i) (fun i => hπ i)).trans h
· intro h
obtain ⟨ρ,hρm,hρ1,hρ,hmass⟩ := hspec hT
apply mass_le_of_finite_tests hT π hπ ρ hρm hρ1 hρ hmass
intro N a ha hbudget
let q : CoefficientList (X := X) (N+1) := ⟨fun i => ⟨a i,(ha i).continuous⟩,ha,hbudget⟩
change (∑ i, T (q.val i) (π i)) ≤ C
apply (hbs N).induction_on (p := fun r : CoefficientList (X := X) (N+1) =>
(∑ i, T (r.val i) (π i)) ≤ C) q
· exact isClosed_le (coefficientList_eval_continuous hT _ (fun i => hπ i)) continuous_const
· exact h N
theorem ae_mass_le_of_listwise {A : Type*} [MeasurableSpace A] (ν : Measure A)
{k : ℕ} {T : A → Functional X k} {C : A → ℝ}
(hT : ∀ᵐ a ∂ν, IsMetricCurrent (T a))
(h : ∀ N (b : Fin (N+1) → X → ℝ) (π : Fin (N+1) → Fin k → X → ℝ),
(∀ i, BoundedLip (b i)) → (∀ x, ∑ i, |b i x| ≤ 1) →
(∀ j i, LipschitzWith 1 (π j i)) →
∀ᵐ a ∂ν, (∑ i, T a (b i) (π i)) ≤ C a) :
∀ᵐ a ∂ν, mass (T a) ≤ C a := by
obtain ⟨π,b,hπ,hb,hbudget,hmass⟩ := exists_countable_mass_lists (X := X) k
have hh : ∀ᵐ a ∂ν, ∀ N j, (∑ i : Fin (N+1), T a (b N j i) (π i)) ≤ C a :=
ae_all_iff.mpr fun N => ae_all_iff.mpr fun j =>
h N (b N j) (fun i => π i) (hb N j) (hbudget N j) (fun i => hπ i)
filter_upwards [hT,hh] with a hta ha
exact (hmass hta (C a)).mpr ha
end BorelCoefficients
end CAT0Fillings
end
section
open Set Filter MeasureTheory Metric
open scoped Topology NNReal ENNReal
namespace CAT0Fillings.SmoothCutoff
lemma dprofile_integrable_measure (μ : Measure ℝ) [IsFiniteMeasure μ] (a : ℝ≥0) (t : ℝ) :
Integrable (dprofile a t) μ := by
obtain ⟨C,hC⟩ := exists_profile_bounds
exact Integrable.of_bound (hC a t).2.1.continuous.aestronglyMeasurable ((C*a:ℝ≥0):ℝ)
(Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using (hC a t).2.2 x)
theorem ae_tendsto_dprofile_measure (μ : Measure ℝ) [IsFiniteMeasure μ] :
∀ᵐ t : ℝ, Tendsto (fun n : ℕ => ∫ x : ℝ,
dprofile ((n:ℝ≥0)+1) t x ∂μ) atTop (𝓝 ((μ.rnDeriv volume t).toReal)) := by
obtain ⟨C,hC⟩ := exists_profile_bounds
let σ := μ.singularPart volume
let f : ℝ → ℝ := fun x => (μ.rnDeriv volume x).toReal
have hf : Integrable f := μ.integrable_toReal_rnDeriv
let a : ℕ → ℝ≥0 := fun n => (n:ℝ≥0)+1
let r : ℕ → ℝ := fun n => ((a n:ℝ))⁻¹
have hap n : 0 < a n := by dsimp [a]; positivity
have harp n : 0 < (a n:ℝ) := hap n
have hrp n : 0 < r n := inv_pos.mpr (harp n)
have hrt : Tendsto r atTop (𝓝[>] 0) := by
apply tendsto_nhdsWithin_iff.mpr
refine ⟨?_,Eventually.of_forall fun n => hrp n⟩
exact tendsto_inv_atTop_zero.comp
(show Tendsto (fun n : ℕ => (a n:ℝ)) atTop atTop from by
simpa only [a,NNReal.coe_add,NNReal.coe_natCast,NNReal.coe_one] using
tendsto_atTop_add_const_right atTop 1 (tendsto_natCast_atTop_atTop (R := ℝ)))
filter_upwards [ae_tendsto_dprofile_integral hf,Besicovitch.ae_tendsto_rnDeriv σ volume,
μ.rnDeriv_singularPart volume] with t ht hσ hz
have hz' : σ.rnDeriv volume t = 0 := hz
have hs : Tendsto (fun n => ∫ x : ℝ, dprofile (a n) t x ∂σ) atTop (𝓝 0) := by
have hratio := (ENNReal.continuousAt_toReal (by simpa only [hz'] using ENNReal.zero_ne_top)).tendsto.comp
(hσ.comp hrt)
have hbound n : ‖∫ x : ℝ, dprofile (a n) t x ∂σ‖ ≤
(2*(C:ℝ)) * (σ (closedBall t (r n))/volume (closedBall t (r n))).toReal := by
have hnz (x : ℝ) (hx : x ∉ closedBall t (r n)) : dprofile (a n) t x = 0 := by
apply dprofile_zero (hap n)
intro hi
apply hx
rw [Real.closedBall_eq_Icc]
change t-r n ≤ x ∧ x ≤ t+r n
exact ⟨by linarith [hi.1,hrp n],hi.2⟩
calc
_ ≤ ∫ x : ℝ, ‖dprofile (a n) t x‖ ∂σ := norm_integral_le_integral_norm _
_ = ∫ x in closedBall t (r n), ‖dprofile (a n) t x‖ ∂σ :=
(setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx => by rw [hnz x hx,norm_zero])).symm
_ ≤ ∫ _x in closedBall t (r n), (C:ℝ)*(a n:ℝ) ∂σ := by
apply integral_mono (dprofile_integrable_measure σ _ _).norm.integrableOn (integrable_const _)
intro x
simpa only [Real.norm_eq_abs,NNReal.coe_mul] using (hC (a n) t).2.2 x
_ = _ := by
rw [integral_const,measureReal_def,Measure.restrict_apply_univ,smul_eq_mul,ENNReal.toReal_div]
change σ.real (closedBall t (r n)) * ((C:ℝ)*(a n:ℝ)) =
(2*(C:ℝ))*(σ.real (closedBall t (r n))/volume.real (closedBall t (r n)))
rw [Real.volume_real_closedBall (hrp n).le]
dsimp only [r]
field_simp
apply squeeze_zero_norm hbound
simpa only [hz',ENNReal.toReal_zero,mul_zero,Function.comp_apply] using hratio.const_mul (2*(C:ℝ))
have heq n : (∫ x : ℝ, dprofile (a n) t x ∂μ) =
(∫ x : ℝ, dprofile (a n) t x ∂σ) +
∫ x : ℝ, dprofile (a n) t x*f x := by
calc
_ = (∫ x : ℝ, dprofile (a n) t x ∂σ) +
∫ x : ℝ, dprofile (a n) t x ∂volume.withDensity (μ.rnDeriv volume) := by
rw [←integral_add_measure (dprofile_integrable_measure σ _ _)
(dprofile_integrable_measure (volume.withDensity (μ.rnDeriv volume)) _ _)]
rw [show σ + volume.withDensity (μ.rnDeriv volume) = μ from μ.singularPart_add_rnDeriv volume]
_ = _ := by
congr 1
rw [integral_withDensity_eq_integral_toReal_smul (μ.measurable_rnDeriv volume)
(μ.rnDeriv_lt_top volume)]
apply integral_congr_ae
exact Eventually.of_forall fun x => mul_comm _ _
change Tendsto (fun n => ∫ x : ℝ, dprofile (a n) t x ∂μ) atTop (𝓝 (f t))
simp_rw [heq]
simpa only [zero_add] using hs.add ht
end CAT0Fillings.SmoothCutoff
end
end OAI
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