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9655094 | 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 | import OAI.Analysis.CircleDomains.Probability.ConfigurationMeasures
import OAI.Analysis.CircleDomains.Selection.AtomicHitTests
namespace OAI
/-! The mean measure of separated unit-square-norm configurations has atomic
square norm at most one. Separation makes the point assignment injective;
Cauchy--Schwarz then passes the finite-dimensional bound through averaging. -/
open Set Function Filter MeasureTheory
open scoped Topology ENNReal
noncomputable section
namespace CircleDomainRigidity
private theorem sum_sq_fibers_le {I X : Type*} [Fintype I] [DecidableEq X]
(p : I → X) (hp : Injective p) (w : I → ℝ) (F : Finset X) :
∑ x ∈ F, (∑ i, if p i = x then w i else 0) ^ 2 ≤ ∑ i, w i ^ 2 := by
classical
have he (x : X) : (∑ i, if p i = x then w i else 0) ^ 2 =
∑ i, if p i = x then w i ^ 2 else 0 := by
by_cases hx : ∃ i, p i = x
· obtain ⟨i, hi⟩ := hx
have heq (j : I) : p j = x ↔ j = i := by
constructor
· intro h; exact hp (h.trans hi.symm)
· rintro rfl; exact hi
simp only [heq, Finset.sum_ite_eq', Finset.mem_univ, ite_true]
· have hn (i : I) : p i ≠ x := fun hi => hx ⟨i, hi⟩
simp [hn]
simp_rw [he]
rw [Finset.sum_comm]
apply Finset.sum_le_sum
intro i _
simp only [Finset.sum_ite_eq]
split
· exact le_rfl
· exact sq_nonneg _
private theorem sum_sq_integrals_le_one {I T : Type*} [Fintype I] [MeasurableSpace T]
(P : ProbabilityMeasure T) (h : I → T → ℝ)
(hi : ∀ i, Integrable (h i) (P : Measure T))
(hbound : ∀ t, ∑ i, h i t ^ 2 ≤ 1) :
∑ i, (∫ t, h i t ∂(P : Measure T)) ^ 2 ≤ 1 := by
let v : I → ℝ := fun i => ∫ t, h i t ∂(P : Measure T)
let S := ∑ i, v i ^ 2
have hS0 : 0 ≤ S := Finset.sum_nonneg fun _ _ => sq_nonneg _
have hb (t : T) : ∑ i, v i * h i t ≤ Real.sqrt S := by
have hh : Real.sqrt (∑ i, h i t ^ 2) ≤ 1 :=
Real.sqrt_le_iff.mpr ⟨by norm_num, by simpa using hbound t⟩
exact (Real.sum_mul_le_sqrt_mul_sqrt Finset.univ v (fun i => h i t)).trans
(by simpa only [S, mul_one] using mul_le_mul_of_nonneg_left hh (Real.sqrt_nonneg S))
have hsi : Integrable (fun t => ∑ i, v i * h i t) (P : Measure T) :=
integrable_finsetSum _ fun i _ => (hi i).const_mul (v i)
have he : ∫ t, ∑ i, v i * h i t ∂(P : Measure T) = S := by
rw [integral_finsetSum _ (fun i _ => (hi i).const_mul (v i))]
simp only [integral_const_mul, S, v, sq]
have hle : S ≤ Real.sqrt S := by
calc
S = ∫ t, ∑ i, v i * h i t ∂(P : Measure T) := he.symm
_ ≤ Real.sqrt S := by
simpa only [integral_const, probReal_univ, one_smul] using
integral_mono_ae hsi (integrable_const (Real.sqrt S)) (Eventually.of_forall hb)
have hsq := Real.sq_sqrt hS0
change S ≤ 1
nlinarith [Real.sqrt_nonneg S]
variable {X : Type*} [MetricSpace X] [SecondCountableTopology X]
[MeasurableSpace X] [BorelSpace X] {n : ℕ} {r : ℝ}
omit [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X] in
theorem HitConfiguration.points_injective (hr : 0 < r) (p : HitConfiguration X n r) :
Injective p.val.1 := by
intro i j hij
by_contra hne
have h := p.property.2 i j hne
rw [hij, dist_self] at h
exact (not_le.mpr hr) h
theorem configurationMeasure_atom_real [DecidableEq X] (p : HitConfiguration X n r) (x : X) :
(configurationMeasure p {x}).toReal =
∑ i, if p.val.1 i = x then (p.val.2 i : ℝ) else 0 := by
classical
simp only [configurationMeasure, Measure.coe_finsetSum, Finset.sum_apply,
Measure.smul_apply, smul_eq_mul, Measure.dirac_apply, Set.indicator_apply,
mem_singleton_iff, Pi.one_apply]
rw [ENNReal.toReal_sum (fun i _ => by split_ifs <;> simp)]
apply Finset.sum_congr rfl
intro i _
split_ifs <;> simp [ENNReal.toReal_ofReal (p.val.2 i).property.1]
theorem configurationMeasure_hasAtomicSquareBound (hr : 0 < r)
(p : HitConfiguration X n r) : HasAtomicSquareBound (configurationMeasure p) := by
classical
intro F
simp_rw [configurationMeasure_atom_real]
exact (sum_sq_fibers_le p.val.1 (HitConfiguration.points_injective hr p) (fun i => (p.val.2 i : ℝ)) F).trans
p.property.1
instance (p : HitConfiguration X n r) : IsFiniteMeasure (configurationMeasure p) :=
⟨lt_of_le_of_lt (configurationMeasure_mass_le p) (by simp)⟩
theorem meanConfigurationMeasure_atom_real
(P : ProbabilityMeasure (HitConfiguration X n r)) (x : X) :
(meanConfigurationMeasure P {x}).toReal =
∫ p, (configurationMeasure p {x}).toReal ∂(P : Measure (HitConfiguration X n r)) := by
rw [meanConfigurationMeasure, Measure.bind_apply (measurableSet_singleton x)
measurable_configurationMeasure.aemeasurable]
exact (integral_toReal
((Measure.measurable_coe (measurableSet_singleton x)).comp
measurable_configurationMeasure).aemeasurable
(Eventually.of_forall fun p => measure_lt_top (configurationMeasure p) {x})).symm
theorem integrable_configuration_atom
(P : ProbabilityMeasure (HitConfiguration X n r)) (x : X) :
Integrable (fun p => (configurationMeasure p {x}).toReal)
(P : Measure (HitConfiguration X n r)) := by
apply Integrable.of_bound
(((Measure.measurable_coe (measurableSet_singleton x)).comp
measurable_configurationMeasure).ennreal_toReal.aestronglyMeasurable) n
apply Eventually.of_forall
intro p
rw [Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]
change (configurationMeasure p {x}).toReal ≤ (n : ℝ)
have hle : configurationMeasure p {x} ≤ n :=
(measure_mono (subset_univ _)).trans (configurationMeasure_mass_le p)
simpa only [ENNReal.toReal_natCast] using ENNReal.toReal_mono (by simp) hle
theorem meanConfigurationMeasure_hasAtomicSquareBound (hr : 0 < r)
(P : ProbabilityMeasure (HitConfiguration X n r)) :
HasAtomicSquareBound (meanConfigurationMeasure P) := by
classical
intro F
let a : F → HitConfiguration X n r → ℝ := fun x p => (configurationMeasure p {x.val}).toReal
have hi (x : F) : Integrable (a x) (P : Measure (HitConfiguration X n r)) :=
integrable_configuration_atom P x.val
have hb (p : HitConfiguration X n r) : ∑ x : F, a x p ^ 2 ≤ 1 := by
exact (Finset.sum_coe_sort F (fun x : X => (configurationMeasure p {x}).toReal ^ 2)).le.trans
(configurationMeasure_hasAtomicSquareBound hr p F)
have h := sum_sq_integrals_le_one P a hi hb
simp_rw [meanConfigurationMeasure_atom_real]
exact (Finset.sum_coe_sort F (fun x : X =>
(∫ p, (configurationMeasure p {x}).toReal ∂(P : Measure (HitConfiguration X n r))) ^ 2)).symm.le.trans h
end CircleDomainRigidity
end
end OAI
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