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a89d282 | 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 | import OAI.Probability.InvariantIsing.Spectral.MeasureInverseContinuity
namespace OAI
/-! Compact-support estimates for the general spectral transform. -/
noncomputable section
open MeasureTheory Filter Set
open scoped Topology Classical
namespace InvariantIsing
lemma integrable_spectralCoordinate (μ : Measure ℝ) [IsProbabilityMeasure μ]
{l e : ℝ} (hμ : ∀ᵐ y ∂μ, y ∈ Icc l e) : Integrable (fun y => y) μ := by
apply (integrable_const (|l| + |e|)).mono' measurable_id.aestronglyMeasurable
filter_upwards [hμ] with y hy
rw [Real.norm_eq_abs, abs_le]
simp only [id_eq]
constructor
· linarith [neg_abs_le l, abs_nonneg e, hy.1]
· linarith [le_abs_self e, abs_nonneg l, hy.2]
theorem measureR_bounds_all (μ : Measure ℝ) [IsProbabilityMeasure μ]
{l e : ℝ} (hμ : ∀ᵐ y ∂μ, y ∈ Icc l e) (x : ℝ) :
l ≤ measureR μ e x ∧ measureR μ e x ≤ e := by
by_cases hx : 0 < x
· exact measureR_bounds μ hμ hx
· rw [measureR, ite_eq_right hx]
constructor
· simpa only [integral_const, probReal_univ, one_smul] using
integral_mono_ae (integrable_const l) (integrable_spectralCoordinate μ hμ)
(hμ.mono fun _ h => h.1)
· simpa only [integral_const, probReal_univ, one_smul] using
integral_mono_ae (integrable_spectralCoordinate μ hμ) (integrable_const e)
(hμ.mono fun _ h => h.2)
theorem measureInverse_uncapped_small (μ : Measure ℝ) [IsProbabilityMeasure μ]
{K e x : ℝ} (hK : 0 < K) (he : e ≤ K)
(hμ : ∀ᵐ y ∂μ, y ∈ Icc (-K) e) (hx : 0 < x) (hsmall : 4 * K * x ≤ 1) :
e < measureInverse μ e x ∧ measureResolvent μ (measureInverse μ e x) = x := by
have htest : e < 1 / x - K := by
have h := (le_div_iff₀ hx).mpr hsmall
nlinarith
have hl := le_measureResolvent μ hμ htest
have hvalue : x ≤ measureResolvent μ (1 / x - K) := by
simpa only [sub_neg_eq_add, sub_add_cancel, one_div_one_div] using hl
have hex := exists_measureResolvent_solution μ (hμ.mono fun _ h => h.2) htest hx hvalue
obtain ⟨b, hb, hsol⟩ := hex
rw [measureInverse_eq_of_solution μ (hμ.mono fun _ h => h.2) hb hsol]
exact ⟨hb, hsol⟩
theorem measureR_sub_mean_identity (μ : Measure ℝ) [IsProbabilityMeasure μ]
{l e x : ℝ} (hμ : ∀ᵐ y ∂μ, y ∈ Icc l e) (hx : 0 < x)
(hs : e < measureInverse μ e x ∧ measureResolvent μ (measureInverse μ e x) = x) :
measureR μ e x - ∫ y, y ∂μ =
∫ y, (measureR μ e x - y) ^ 2 / (measureInverse μ e x - y) ∂μ := by
let b := measureInverse μ e x
have hi := integrable_spectralCoordinate μ hμ
have hk : Integrable (fun y => 1 / (b - y)) μ :=
integrable_resolventKernel μ (hμ.mono fun _ h => h.2) hs.1
have ha : ∀ᵐ y ∂μ, (measureR μ e x - y) ^ 2 / (b - y) =
(b - y) - 2 / x + (1 / x ^ 2) * (1 / (b - y)) := by
filter_upwards [hμ] with y hy
have hby : b - y ≠ 0 := by dsimp only [b]; linarith [hs.1, hy.2]
simp only [measureR, ite_eq_left hx]
change (b - 1 / x - y) ^ 2 / (b - y) = _
field_simp [hx.ne', hby]
ring
have hibase : Integrable (fun y => b - y) μ := (integrable_const b).sub hi
have hib : Integrable (fun y => b - y - 2 / x) μ :=
hibase.sub (integrable_const (2 / x))
have hir : Integrable (fun y => (1 / x ^ 2) * (1 / (b - y))) μ := hk.const_mul _
have hsplit : (∫ y, b - y - 2 / x ∂μ) = b - (∫ y, y ∂μ) - 2 / x := by
calc
_ = (∫ y, b - y ∂μ) - (∫ _, 2 / x ∂μ) :=
integral_sub hibase (integrable_const (2 / x))
_ = _ := by
rw [show (∫ y, b - y ∂μ) = (∫ _, b ∂μ) - (∫ y, y ∂μ) from
integral_sub (integrable_const b) hi]
simp
rw [integral_congr_ae ha, integral_add hib hir, hsplit, integral_const_mul]
change measureR μ e x - (∫ y, y ∂μ) =
b - (∫ y, y ∂μ) - 2 / x + (1 / x ^ 2) * measureResolvent μ b
rw [hs.2]
simp only [measureR, ite_eq_left hx]
change b - 1 / x - (∫ y, y ∂μ) = _
field_simp
ring
theorem measureR_sub_mean_bound (μ : Measure ℝ) [IsProbabilityMeasure μ]
{K e x : ℝ} (hK : 0 < K) (he : e ≤ K)
(hμ : ∀ᵐ y ∂μ, y ∈ Icc (-K) e) (hx : 0 < x) (hsmall : 4 * K * x ≤ 1) :
0 ≤ measureR μ e x - ∫ y, y ∂μ ∧
measureR μ e x - ∫ y, y ∂μ ≤ 8 * K ^ 2 * x := by
let b := measureInverse μ e x
let r := measureR μ e x
have hs := measureInverse_uncapped_small μ hK he hμ hx hsmall
have hr := measureR_bounds μ hμ hx
have hbr : x * b = x * r + 1 := by
dsimp only [b, r]
simp only [measureR, ite_eq_left hx]
field_simp
ring
have hb : ∀ᵐ y ∂μ, 0 ≤ (r - y) ^ 2 / (b - y) ∧
(r - y) ^ 2 / (b - y) ≤ 8 * K ^ 2 * x := by
filter_upwards [hμ] with y hy
have hden : 0 < b - y := by dsimp only [b]; linarith [hs.1, hy.2]
have hlo : -2 * K ≤ r - y := by dsimp only [r]; linarith [hr.1, hy.2]
have hhi : r - y ≤ 2 * K := by dsimp only [r]; linarith [hr.2, hy.1]
have hsq : (r - y) ^ 2 ≤ 4 * K ^ 2 := by
nlinarith [mul_nonneg (sub_nonneg.mpr hlo) (sub_nonneg.mpr hhi)]
have hinv : 1 / (b - y) ≤ 2 * x := by
apply (div_le_iff₀ hden).mpr
have := mul_le_mul_of_nonneg_left hlo hx.le
nlinarith
refine ⟨div_nonneg (sq_nonneg _) hden.le, ?_⟩
calc
(r - y) ^ 2 / (b - y) = (r - y) ^ 2 * (1 / (b - y)) := by ring
_ ≤ (4 * K ^ 2) * (2 * x) := mul_le_mul hsq hinv (one_div_pos.mpr hden).le (by positivity)
_ = 8 * K ^ 2 * x := by ring
have hm : Measurable (fun y : ℝ => (r - y) ^ 2 / (b - y)) :=
((measurable_const.sub measurable_id).pow_const 2).div (measurable_const.sub measurable_id)
have hi : Integrable (fun y => (r - y) ^ 2 / (b - y)) μ := by
apply (integrable_const (8 * K ^ 2 * x)).mono' hm.aestronglyMeasurable
filter_upwards [hb] with y hy
simpa only [Real.norm_eq_abs, abs_of_nonneg hy.1] using hy.2
rw [measureR_sub_mean_identity μ hμ hx hs]
constructor
· exact integral_nonneg_of_ae (hb.mono fun _ h => h.1)
· have hu := integral_mono_ae hi (integrable_const (8 * K ^ 2 * x)) (hb.mono fun _ h => h.2)
simpa only [integral_const, probReal_univ, one_smul] using hu
theorem continuousAt_measureR_zero (μ : Measure ℝ) [IsProbabilityMeasure μ]
{K e : ℝ} (hK : 0 < K) (he : e ≤ K) (hμ : ∀ᵐ y ∂μ, y ∈ Icc (-K) e) :
ContinuousAt (measureR μ e) 0 := by
have herr : ∀ᶠ x : ℝ in 𝓝 0,
‖measureR μ e x - ∫ y, y ∂μ‖ ≤ 8 * K ^ 2 * |x| := by
filter_upwards [Iio_mem_nhds (show (0 : ℝ) < 1 / (4 * K) by positivity)] with x hxsmall
by_cases hx : 0 < x
· have hsmall : 4 * K * x ≤ 1 := by
have := (lt_div_iff₀ (show (0 : ℝ) < 4 * K by positivity)).mp hxsmall
nlinarith
have hb := measureR_sub_mean_bound μ hK he hμ hx hsmall
simpa only [Real.norm_eq_abs, abs_of_nonneg hb.1, abs_of_pos hx] using hb.2
· simp only [measureR, ite_eq_right hx, sub_self, norm_zero]
positivity
have ht : Tendsto (fun x : ℝ => 8 * K ^ 2 * |x|) (𝓝 0) (𝓝 0) := by
simpa using (continuous_abs.tendsto (0 : ℝ)).const_mul (8 * K ^ 2)
have hl := (tendsto_sub_nhds_zero_iff).mp (squeeze_zero_norm' herr ht)
simpa only [ContinuousAt, measureR, lt_self_iff_false, ite_false] using hl
theorem continuous_measureR (μ : Measure ℝ) [IsProbabilityMeasure μ]
{K e : ℝ} (hK : 0 < K) (he : e ≤ K) (hμ : ∀ᵐ y ∂μ, y ∈ Icc (-K) e) :
Continuous (measureR μ e) := by
apply continuous_iff_continuousAt.mpr
intro x
rcases lt_trichotomy x 0 with hx | hx | hx
· have hc : ContinuousAt (fun _ : ℝ => ∫ y, y ∂μ) x := continuousAt_const
apply hc.congr_of_eventuallyEq
filter_upwards [Iio_mem_nhds hx] with y hy
simp only [measureR, ite_eq_right (not_lt.mpr (show y < 0 from hy).le)]
· subst x
exact continuousAt_measureR_zero μ hK he hμ
· exact continuousAt_measureR_pos μ (hμ.mono fun _ h => h.2) hx
end InvariantIsing
end
end OAI
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