Cambridge University Press 0521615259 - Measures, Integrals and Martingales Rene L. Schilling Index More information Notation index This is intended to aid cross-referencing, so notation that is specific to a single sec- tion is generally not listed. Some symbols are used locally, without ambiguity, in senses other than those given below. Numbers following entries are page numbers with the occasional (Pr mn) referring to Problem mn on the respective page. Unless otherwise stated, binary operations between functions such as f ± g, f · g, → ∧ ∨ −−→j f g, f g, comparisons f g, f<gor limiting relations fj f, limj fj, lim infj fj, lim sup fj, supi fi or infi fi are always understood pointwise. Alternatives are indicated by square brackets, i.e., ‘if A [B]…then P [Q] ’ should be read as ‘if A … then P ’ and ‘if B … then Q ’. Abbreviations and shorthand notation a.a. almost all, 80 ∪-stable stable under finite unions a.e. almost every(where), 80 ∩-stable stable under finite ONB orthonormal basis, 239 intersections, 32 ONS orthonormal system, 239 end of proof, x UI uniformly integrable, 163, [ ] indicates that a small 194 intermediate step is w.r.t. with respect to required, x negative always in the sense 0 (in the margin) caution, x positive always in the sense 0 Special labels, defining properties 1 2 3 Dynkin system, 31 S1 S2 S3 semi-ring, 37 M1 M2 measure, 22 1 2 3-algebra, 15 Mathematical symbols Sub- and superscripts ⊥ orthogonal complement, 235 + positive part, b bounded positive elements c compact support 367 © Cambridge University Press www.cambridge.org Cambridge University Press 0521615259 - Measures, Integrals and Martingales Rene L. Schilling Index More information 368 Notation index Symbols, binary operations #A #B,#A<#B 7 t · A = taa∈ A, 36 (Pr 5.8) ∀ x + A = x + aa∈ A,28 for all, for every ∩ = ∩ ∈ ∃ there exists, there is E E AA ,16 # cardinality, 7 −→ converges to a b a b a b a b open, closed, half-open intervals −→ convergence in measure, ab, ((a,b)) rectangles in n,18 163 ↑ increases to u v u = v u ∈ B etc. 57 ↓ decreases to = defining equality Functions, norms, measures & integrals def = equal by definition ≡ identically equal fA = fx x ∈ A,6 ∨,[f ∨ g] maximum [of f and g], 64 f −1 = f −1BB∈ ,16 ∧,[f ∧ g] minimum [of f and g], 64 f g composition: absolutely continuous, 202 f gx = fgx ⊥ - measures: singular, 209 f + = f ∨ 0 positive part, 61 - Hilbert space: orthogonal, f − =−f ∧ 0 negative part, 231, 235 61 convolution, 137 1A indicator function of A ⊕ direct sum, 236 1x∈ A × 1 x = - Cartesian product of sets; A 0x ∈ A - Cartesian product of sgn sign function -algebras, 121; ⎧ ⎨1x>0 - product of measures, 125 = = ⊗ product of -algebras, 121; sgnx ⎩0x0 −1x<0 n Set operations • maximum-norm in and n×n, 142 ∅ p empty set • p L -norm, 105, 108 ∪ A B union 5 • L -norm, 116 ∪· A B union of disjoint sets, 3, 5 • • scalar product, 228 A ∩ B intersection, 5 A \ B set-theoretic difference, 5 ¯ completion of the measure Ac complement of A,5 , 29 (Pr 4.13) A B symmetric difference, 13 , restriction of the measure (Pr 2.2) to the family of sets ⊂ A B subset, 5 X, restriction of the measure X A B proper subset, 5 to the canonical -algebra A¯ closure of A, 320 on X A open interior of A, 320 T −1 image measure, 51 ↑ Aj A 24 - lim → convergence in measure, ↓ j Aj A 24 163 A × B Cartesian product T image measure, 51 An n-fold Cartesian product · u measure with density, 79–80 A infinite sequences with values in A ud, ux dx , ux dx 69, 76 #A cardinality of A,7 = A ud 1Aud ,79 © Cambridge University Press www.cambridge.org Cambridge University Press 0521615259 - Measures, Integrals and Martingales Rene L. Schilling Index More information Notation index 369 udx, ux dx 77 GLn invertible n×n-matrices = u dT u Td , 134 b b id identity map or matrix a ux dx, R a ux dx Riemann integral, 93, 339 n n half-open rectangles in n, 18 Other notation in alphabetical order n rat rat …with rational endpoints, ℵ 18 0 cardinality of ,7 o on o n n ∗ open rectangles in ,18 completion, 29 (Pr 4.13) o on rat rat …with rational endpoints, jj, filtration, 176 = ∈ 18 i i I, 177, 203 = − ∈− , 193 or , 185 n n B x open ball with radius r and Lebesgue measure in ,27 r 1 centre x, 17, 323 79 p A Borel sets in A, 20 (Pr 3.10) 113 1 1 Borel sets in , 226 , ¯ 76 n n n 1 Borel sets in ,17 227 ∗ n 1 completion of the Borel sets, -lim∈I 203 132 (Pr 13.11), 144, 330 Lp 108 ¯ p p ¯ Borel sets in ¯ ,58 , L 228 p 105 p cardinality of 0 1,11 -limj→ 109 complex numbers , L 116 L 260 CU continuous functions = fU→ lim infj aj supk infjk aj, 313 lim sup a = inf sup a , 313 CcU continuous functions j j k j k j → = fU with compact lim infj Aj k∈ jk Aj, 316 support = lim supj Aj k∈ jk Aj , 316 CU functions fU→ differentiable arbitrarily , ¯ 59 often M 258 Dynkin system generated by natural numbers: 1 2 3 ,31 0 positive integers: 0 1 2 unit mass at x, Dirac x -null sets, 29 (Pr 4.10), 80 measure at x,26 det determinant (of a matrix) n volume of the unit ball in Dx Jacobian, 147 n, 156 d d Radon-Nikodýmderivative, X topology, open sets, 17 203 n, n topology, open sets in n, 17 E , E• conditional expectation, 250, 263 PC , PF (orthogonal) projection, 235 E = E conditional X all subsets of X,12 expectation, 260, 263 simple functions, 60 rational numbers © Cambridge University Press www.cambridge.org Cambridge University Press 0521615259 - Measures, Integrals and Martingales Rene L. Schilling Index More information 370 Notation index ∈ real numbers T Ti i I -algebra generated by ¯ extended real line the map(s) T, resp., Ti,51 − +,58 span all finite linear combinations n Euclidean n-space of the elements in , 239 n n supp f = f = 0 support of f x, y 147 n×n real n × n-matrices a b 339 , stopping times, 185 = − a , −b 354 x shift xy y x,49 a b, a b 358 a b, − 359 X measurable space, 22 X measure space, 22 X j, X filtered , stopping times, 185 measure space, 176, 203 -algebra generated by , 16 integers: 0 ±1 ±2 © Cambridge University Press www.cambridge.org Cambridge University Press 0521615259 - Measures, Integrals and Martingales Rene L. Schilling Index More information Name and subject index This should be used in conjunction with the Bibliography and the Index of Notation. Numbers following entries are page numbers which, if accomplished by (Pr n.m), refer to Problem n.m on that page; a number with a trailing ‘n’ indicates that a footnote is being referenced. Unless otherwise started ‘integral’, integrability’ etc. always refer to the (abstract) Lebesgue integral. Within the index we use ‘L-…’ and ‘R-…’ as a shorthand for ‘(abstract) Lebesgue-…’ and ‘Riemann-…’ ℵ Bernstein polynomials, 280 0 aleph null,7 absolutely continuous, 202 bijective map, 6 uniformly absolutely continous, 169 Boas, Ralph, 114 Alexits, Gyorgy,¨ 277n, 302 Borel, Emile almost all (a.a.), 80 Borel measurable, 17 almost everywhere (a.e.), 80 Borel set, 17 Analytic set, 333 Borel -algebra, 17 Andrews, George, 277n alternative definition, 21 (Pr 3.12) arc-length, 160–161 (Pr 15.6) cardinality, 332 Askey, Richard, 277n completion, 330 atom, 20 (Pr 3.5), 46 (Pr 6.5) generator of, 18, 19 axiom of choice, 331 in a subset, 20 (Pr 3.10) in ¯ ,58 Banach, Stephan, 43 Brownian motion, 309–311 Banach space, 326 Banach–Tarski paradox, 43 continuum,11 basic convergence result Calderón, Alberto for improper R-integrals, 355 Calderón–Zygmund decomposition, 221 for R-integrals, 351 Cantor, Georg, 11 basis, 242 Cantor’s diagonal method, 11 unconditional basis, 293–295 Cantor discontinuum, 55 (Pr 7.10), Bass, Richard, 311 223–224 (Pr 19.10) Bauer, Heinz, 159, 281, 310 Cantor function, 224 (Pr 19.10) Benyamini, Yoav, 210 Cantor (ternary) set, 55 (Pr 7.10), Bernoulli distribution, 183 223–224 (Pr 19.10) Bernstein, Serge˘ı N., 279 Carathéodory, Constantin, 37 371 © Cambridge University Press www.cambridge.org Cambridge University Press 0521615259 - Measures, Integrals and Martingales Rene L. Schilling Index More information 372 Name and Subject Index cardinality, 7 conditional expectation of the Borel -algebra, 332 in Lp and L , 260 of the Lebesgue -algebra, 330 in L1, 263–264 Carleson, Lennart, 289 in L2, 250 Cartesian product properties (in L2), 251 rules for Cartesian Products, 121 properties (in Lp), 261–262 Cauchy sequence via Radon-Nikodým Theorem, in p, 109 223 (Pr 19.3) in metric spaces, 325 conjugate numbers (also conj. indices), 105 in normed spaces, 234 conjugate Young functions, 117 (Pr 12.5) Cavalieri’s principle, 120 continuity Cesàro mean, 286 implies measurability, 50 ∅ change of variable formula of measures at ,24 for Lebesgue integrals, 151 of measures from above, 24 for Riemann integrals, 350 of measures from below, 24 for Stieltjes integrals, 133 (Pr 13.13) in metric spaces, 324 Chebyshev, Pafnuti L., 85 (Pr 10.5) in topological spaces, 321 Chebyshev polynomials (first kind), 277 continuous function Ciesielski, Z., 311 is measurable, 50 closed ball, 323 is Riemann integrable, 342 compactness (weak sequential), 169 continuous linear functional in Hilbert space, 238 in 1, 168 representation of continuous linear in p, 168, 274 (Pr 23.8) functionals, 239 and uniform integrability, 169 convergence completeness along an upwards filtering set, 203 of p,1 p<, 110 criteria for a.e.