File size: 19,125 Bytes
7f37411 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphDegrees
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace Attempt4Near
open Set MeasureTheory
open scoped ComplexConjugate
variable {V α β κ : Type*} [MeasurableSpace V]
noncomputable def graphChildSquareND (A : Finset α) (B : Finset β)
(U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ) (r : ℝ) (z : V × V) : ℝ :=
(∑ i ∈ A, ‖U i (r,z.1)‖^2)*(∑ j ∈ B, ‖W j (r,z.2)‖^2)
lemma graphChildSquareND_integrable (μ : Measure V) [IsFiniteMeasure μ]
(A : Finset α) (B : Finset β) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j))
(C : α → ℝ) (D : β → ℝ)
(hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i)
(hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ D j) (r : ℝ) :
Integrable (graphChildSquareND A B U W r) (μ.prod μ) := by
unfold graphChildSquareND
apply Integrable.mul_prod (f := fun w => ∑ i ∈ A, ‖U i (r,w)‖^2)
(g := fun w => ∑ j ∈ B, ‖W j (r,w)‖^2)
· exact integrable_finsetSum A (fun i hi => angular_square_integrableND μ
((hU i hi).comp (measurable_const.prodMk measurable_id)) (hbU i hi r))
· exact integrable_finsetSum B (fun j hj => angular_square_integrableND μ
((hW j hj).comp (measurable_const.prodMk measurable_id)) (hbW j hj r))
lemma graphChildSquareND_integral (μ : Measure V) [IsFiniteMeasure μ]
(A : Finset α) (B : Finset β) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j))
(C : α → ℝ) (D : β → ℝ)
(hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i)
(hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ D j) (r : ℝ) :
(∫ z, graphChildSquareND A B U W r z ∂μ.prod μ) =
∑ i ∈ A, ∑ j ∈ B, sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r := by
dsimp [graphChildSquareND]
rw [integral_prod_mul (fun w => ∑ i ∈ A, ‖U i (r,w)‖^2)
(fun w => ∑ j ∈ B, ‖W j (r,w)‖^2),integral_finsetSum A (f := fun i w => ‖U i (r,w)‖^2) (fun i hi => angular_square_integrableND μ
((hU i hi).comp (measurable_const.prodMk measurable_id)) (hbU i hi r)),
integral_finsetSum B (f := fun j w => ‖W j (r,w)‖^2) (fun j hj => angular_square_integrableND μ
((hW j hj).comp (measurable_const.prodMk measurable_id)) (hbW j hj r))]
exact Finset.sum_mul_sum A B _ _
noncomputable def graphRetainedCrossedND (A : Finset α) (B : Finset β) (K : Finset κ)
(G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)]
(η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (z : V × V) : ℂ :=
∑ k ∈ K, (η k z:ℂ)*(∑ a ∈ A.product B, ∑ b ∈ A.product B,
if G k a b then (U a.1 (r,z.1)*W a.2 (r,z.2))*
conj (U b.1 (r,z.2)*W b.2 (r,z.1)) else 0)
lemma graphRetainedCrossedND_angular_bound (μ : Measure V) [IsFiniteMeasure μ]
(A : Finset α) (B : Finset β) (K : Finset κ)
(G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)]
(η : κ → V × V → ℝ)
(hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1)
(U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j))
(C : α → ℝ) (E : β → ℝ)
(hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i)
(hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) (D : ℝ)
(hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D)
(hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D)
(r : ℝ) :
‖∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ‖ ≤
D*∑ i ∈ A, ∑ j ∈ B, sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r := by
have hi := graphChildSquareND_integrable μ A B U W hU hW C E hbU hbW r
have hb := norm_integral_le_of_norm_le ((hi.add hi.swap).const_mul (D/2))
(show ∀ᵐ z ∂μ.prod μ, ‖graphRetainedCrossedND A B K G η U W r z‖ ≤
D/2*(graphChildSquareND A B U W r z+graphChildSquareND A B U W r z.swap) from by
filter_upwards [hpart] with z hz
exact graph_partition_product_schurND A B K G
(fun i => U i (r,z.1)) (fun i => U i (r,z.2))
(fun j => W j (r,z.2)) (fun j => W j (r,z.1)) (fun k => η k z) hz.1 hz.2 D hrow hcol)
simp only [Pi.add_apply,Function.comp_apply] at hb
rw [integral_const_mul] at hb
have hadd := integral_add hi hi.swap
simp only [Function.comp_apply] at hadd
rw [hadd,integral_prod_swap] at hb
rw [graphChildSquareND_integral μ A B U W hU hW C E hbU hbW] at hb
convert hb using 1
ring
theorem graphRetainedCrossedND_integral_bound (μ : Measure V) [IsFiniteMeasure μ]
(A : Finset α) (B : Finset β) (K : Finset κ)
(G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)]
(η : κ → V × V → ℝ)
(hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1)
(U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j))
(C : α → ℝ) (E : β → ℝ)
(hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i)
(hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) (D : ℝ)
(hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D)
(hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D)
{Ψ : ℝ → ℝ} (hΨ : Integrable Ψ) (hΨpos : ∀ r, 0 ≤ Ψ r) :
‖∫ r, (Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ‖ ≤
D*∑ i ∈ A, ∑ j ∈ B, ∫ r, Ψ r*sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r := by
have hi (i) (hi : i ∈ A) (j) (hj : j ∈ B) :=
graph_weighted_moments_integrableND μ (hU i hi) (hW j hj) (hbU i hi) (hbW j hj) hΨ
have hs := (integrable_finsetSum A (fun i hiA => integrable_finsetSum B
(fun j hjB => hi i hiA j hjB))).const_mul D
have hh := norm_integral_le_of_norm_le hs
(show ∀ᵐ r : ℝ, ‖(Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ‖ ≤
D*∑ i ∈ A, ∑ j ∈ B, Ψ r*sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r from by
filter_upwards with r
rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hΨpos r)]
have hp := mul_le_mul_of_nonneg_left (graphRetainedCrossedND_angular_bound
μ A B K G η hpart U W hU hW C E hbU hbW D hrow hcol r) (hΨpos r)
simpa only [Finset.mul_sum,mul_assoc,mul_left_comm,mul_comm] using hp)
rw [integral_const_mul,integral_finsetSum A (fun i hiA => integrable_finsetSum B
(fun j hjB => hi i hiA j hjB))] at hh
apply hh.trans_eq
congr 1
apply Finset.sum_congr rfl
intro i hiA
exact integral_finsetSum B (fun j hjB => hi i hiA j hjB)
end Attempt4Near
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace Attempt4Near
open Set MeasureTheory
open scoped ComplexConjugate
variable {V α β κ : Type*} [MeasurableSpace V]
noncomputable def graphRetainedDirectND (A : Finset α) (B : Finset β) (K : Finset κ)
(G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)]
(η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (z : V × V) : ℂ :=
∑ k ∈ K, (η k z:ℂ)*(∑ a ∈ A.product B, ∑ b ∈ A.product B,
if G k a b then (U a.1 (r,z.1)*W a.2 (r,z.2))*
conj (U b.1 (r,z.1)*W b.2 (r,z.2)) else 0)
lemma graphRetainedDirectND_angular_bound (μ : Measure V) [IsFiniteMeasure μ]
(A : Finset α) (B : Finset β) (K : Finset κ)
(G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)]
(η : κ → V × V → ℝ)
(hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1)
(U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j))
(C : α → ℝ) (E : β → ℝ)
(hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i)
(hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) (D : ℝ)
(hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D)
(hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D)
(r : ℝ) :
‖∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ‖ ≤
D*∑ i ∈ A, ∑ j ∈ B, sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r := by
have hi := graphChildSquareND_integrable μ A B U W hU hW C E hbU hbW r
have hb := norm_integral_le_of_norm_le (hi.const_mul D)
(show ∀ᵐ z ∂μ.prod μ, ‖graphRetainedDirectND A B K G η U W r z‖ ≤
D*graphChildSquareND A B U W r z from by
filter_upwards [hpart] with z hz
have hh := graph_partition_product_schurND A B K G
(fun i => U i (r,z.1)) (fun i => U i (r,z.1))
(fun j => W j (r,z.2)) (fun j => W j (r,z.2)) (fun k => η k z) hz.1 hz.2 D hrow hcol
convert hh using 1
· rfl
· dsimp [graphChildSquareND]
ring)
rw [integral_const_mul,graphChildSquareND_integral μ A B U W hU hW C E hbU hbW] at hb
exact hb
theorem graphRetainedDirectND_integral_bound (μ : Measure V) [IsFiniteMeasure μ]
(A : Finset α) (B : Finset β) (K : Finset κ)
(G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)]
(η : κ → V × V → ℝ)
(hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1)
(U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j))
(C : α → ℝ) (E : β → ℝ)
(hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i)
(hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) (D : ℝ)
(hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D)
(hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D)
{Ψ : ℝ → ℝ} (hΨ : Integrable Ψ) (hΨpos : ∀ r, 0 ≤ Ψ r) :
‖∫ r, (Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ‖ ≤
D*∑ i ∈ A, ∑ j ∈ B, ∫ r, Ψ r*sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r := by
have hi (i) (hi : i ∈ A) (j) (hj : j ∈ B) :=
graph_weighted_moments_integrableND μ (hU i hi) (hW j hj) (hbU i hi) (hbW j hj) hΨ
have hs := (integrable_finsetSum A (fun i hiA => integrable_finsetSum B
(fun j hjB => hi i hiA j hjB))).const_mul D
have hh := norm_integral_le_of_norm_le hs
(show ∀ᵐ r : ℝ, ‖(Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ‖ ≤
D*∑ i ∈ A, ∑ j ∈ B, Ψ r*sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r from by
filter_upwards with r
rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hΨpos r)]
have hp := mul_le_mul_of_nonneg_left (graphRetainedDirectND_angular_bound
μ A B K G η hpart U W hU hW C E hbU hbW D hrow hcol r) (hΨpos r)
simpa only [Finset.mul_sum,mul_assoc,mul_left_comm,mul_comm] using hp)
rw [integral_const_mul,integral_finsetSum A (fun i hiA => integrable_finsetSum B
(fun j hjB => hi i hiA j hjB))] at hh
apply hh.trans_eq
congr 1
apply Finset.sum_congr rfl
intro i hiA
exact integral_finsetSum B (fun j hjB => hi i hiA j hjB)
end Attempt4Near
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace Attempt4Near
open Set MeasureTheory
open scoped ComplexConjugate
variable {V α β κ : Type*} [MeasurableSpace V]
lemma graph_four_sumND (A : Finset α) (B : Finset β)
(U₁ U₂ : α → ℂ) (W₁ W₂ : β → ℂ) :
(∑ a ∈ A.product B, ∑ b ∈ A.product B,
(U₁ a.1*W₁ a.2)*conj (U₂ b.1*W₂ b.2)) =
((∑ i ∈ A, U₁ i)*(∑ j ∈ B, W₁ j))*conj ((∑ i ∈ A, U₂ i)*(∑ j ∈ B, W₂ j)) := by
rw [← Finset.sum_mul_sum,← map_sum]
simp_rw [Finset.product_eq_sprod,Finset.sum_product,← Finset.sum_mul_sum]
omit [MeasurableSpace V] in
lemma graphRetainedDirectND_complete (A : Finset α) (B : Finset β) (K : Finset κ)
(η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (z : V × V) (hpart : ∑ k ∈ K, η k z=1) :
graphRetainedDirectND A B K (fun _ _ _ => True) η U W r z =
((‖∑ i ∈ A, U i (r,z.1)‖^2*‖∑ j ∈ B, W j (r,z.2)‖^2:ℝ):ℂ) := by
simp only [graphRetainedDirectND,ite_true]
rw [← Finset.sum_mul,← Complex.ofReal_sum,hpart,Complex.ofReal_one,one_mul,
graph_four_sumND A B (fun i => U i (r,z.1)) (fun i => U i (r,z.1))
(fun j => W j (r,z.2)) (fun j => W j (r,z.2)),
Complex.mul_conj,Complex.normSq_eq_norm_sq,norm_mul,mul_pow]
omit [MeasurableSpace V] in
lemma graphRetainedCrossedND_complete (A : Finset α) (B : Finset β) (K : Finset κ)
(η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (z : V × V) (hpart : ∑ k ∈ K, η k z=1) :
graphRetainedCrossedND A B K (fun _ _ _ => True) η U W r z =
((∑ i ∈ A, U i (r,z.1))*conj (∑ j ∈ B, W j (r,z.1)))*
conj ((∑ i ∈ A, U i (r,z.2))*conj (∑ j ∈ B, W j (r,z.2))) := by
simp only [graphRetainedCrossedND,ite_true]
rw [← Finset.sum_mul,← Complex.ofReal_sum,hpart,Complex.ofReal_one,one_mul]
rw [graph_four_sumND A B (fun i => U i (r,z.1)) (fun i => U i (r,z.2))
(fun j => W j (r,z.2)) (fun j => W j (r,z.1))]
simp only [map_mul,starRingEnd_self_apply]
ring
lemma graphExpandedDirectND_integral (μ : Measure V) [SFinite μ]
(A : Finset α) (B : Finset β) (K : Finset κ)
(η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (hpart : ∀ᵐ z ∂μ.prod μ, ∑ k ∈ K, η k z=1) :
(∫ z, graphRetainedDirectND A B K (fun _ _ _ => True) η U W r z ∂μ.prod μ) =
((sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r*
sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r:ℝ):ℂ) := by
calc
_ = ∫ z : V × V, ((‖∑ i ∈ A, U i (r,z.1)‖^2*‖∑ j ∈ B, W j (r,z.2)‖^2:ℝ):ℂ)
∂μ.prod μ := integral_congr_ae (by
filter_upwards [hpart] with z hz
exact graphRetainedDirectND_complete A B K η U W r z hz)
_ = _ := by
rw [integral_complex_ofReal,integral_prod_mul
(fun w => ‖∑ i ∈ A, U i (r,w)‖^2) (fun w => ‖∑ j ∈ B, W j (r,w)‖^2)]
rfl
lemma graphExpandedCrossedND_integral (μ : Measure V) [SFinite μ]
(A : Finset α) (B : Finset β) (K : Finset κ)
(η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (hpart : ∀ᵐ z ∂μ.prod μ, ∑ k ∈ K, η k z=1) :
(∫ z, graphRetainedCrossedND A B K (fun _ _ _ => True) η U W r z ∂μ.prod μ) =
((‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2:ℝ):ℂ) := by
calc
_ = ∫ z : V × V, ((∑ i ∈ A, U i (r,z.1))*conj (∑ j ∈ B, W j (r,z.1)))*
conj ((∑ i ∈ A, U i (r,z.2))*conj (∑ j ∈ B, W j (r,z.2))) ∂μ.prod μ :=
integral_congr_ae (by
filter_upwards [hpart] with z hz
exact graphRetainedCrossedND_complete A B K η U W r z hz)
_ = _ := by
rw [integral_prod_mul
(fun w => (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)))
(fun w => conj ((∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)))),
integral_conj,Complex.mul_conj,Complex.normSq_eq_norm_sq]
end Attempt4Near
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace Attempt4Near
open MeasureTheory
open scoped BigOperators
variable {Z α β κ : Type*} [MeasurableSpace Z]
lemma graph_integral_sum_truncateND (μ : Measure Z) (S : Finset α)
(G : α → Prop) [DecidablePred G] (F : α → Z → ℂ)
(hi : ∀ i ∈ S, Integrable (F i) μ) {ε : ℝ} (hε : 0 ≤ ε)
(hne : ∀ i ∈ S, ¬G i → ‖∫ z, F i z ∂μ‖ ≤ ε) :
‖(∫ z, ∑ i ∈ S, F i z ∂μ)-(∫ z, ∑ i ∈ S, if G i then F i z else 0 ∂μ)‖ ≤
S.card*ε := by
have hit (i : α) (hiS : i ∈ S) : Integrable (fun z => if G i then F i z else 0) μ := by
by_cases h : G i
· simpa only [h,ite_true] using hi i hiS
· simp only [h,ite_false]; exact integrable_zero _ _ _
rw [integral_finsetSum S hi,integral_finsetSum S hit]
have he : (∑ i ∈ S, ∫ z, if G i then F i z else 0 ∂μ) =
∑ i ∈ S.filter G, ∫ z, F i z ∂μ := by
rw [Finset.sum_filter]
apply Finset.sum_congr rfl
intro i _
by_cases h : G i <;> simp [h]
rw [he]
exact graph_discarded_sumND S G (fun i => ∫ z, F i z ∂μ) hε hne
lemma graph_integral_localizationND (μ : Measure Z) (A : Finset α) (B : Finset β) (K : Finset κ)
(G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)]
(F : κ → (α × β) → (α × β) → Z → ℂ)
(hi : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, Integrable (F k a b) μ)
{ε : ℝ} (hε : 0 ≤ ε)
(hne : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B,
¬G k a b → ‖∫ z, F k a b z ∂μ‖ ≤ ε) :
‖(∫ z, ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B, F k a b z ∂μ)-
(∫ z, ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B,
if G k a b then F k a b z else 0 ∂μ)‖ ≤
(K.card:ℝ)*((A.card:ℝ)*B.card)^2*ε := by
let S := K.product ((A.product B).product (A.product B))
have h := graph_integral_sum_truncateND μ S
(fun z => G z.1 z.2.1 z.2.2) (fun z => F z.1 z.2.1 z.2.2)
(fun z hz => hi z.1 (Finset.mem_product.mp hz).1
z.2.1 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).1
z.2.2 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).2) hε
(fun z hz => hne z.1 (Finset.mem_product.mp hz).1
z.2.1 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).1
z.2.2 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).2)
simpa only [S,Finset.product_eq_sprod,Finset.sum_product,Finset.card_product,
Nat.cast_mul,pow_two,mul_assoc] using h
end Attempt4Near
end
end
end OAI
|