Notation General Notation Description See a b & a b The minimum and the maximum of a and b a + & a f S u The non-negative part, a 0, and non-positive part, (a 0) of a R The restriction of the function f to the set S The uniform (supremum) norm Var(ψ; J) Variation norm of function ψ on interval J 1.2.2 Γ(t) Euler s gamma function Ex. 5.1.13 i The imaginary number 1 ω N 1 The surface area of unit sphere S N 1 in R N Ex. 5.2.20 Ω N O(g(t)) o(g(t)) The volume, N 1 ω N 1, of the unit ball B(0;1) in R N A function f for which f(t) is bounded as t tends to a g(t) limit A function f for which f(t) tends to 0 as t tends to a g(t) limit. Sets, Functions, and Spaces t A The integer part of t R The complement of the set A A (δ) The δ-hull around the set A 5.2.1 Γ(ρ) The tubular neighborhood of Γ (5.2.8) 1 A The indicator function of the set A 3.1.1 B E (a,r) The ball of radius r around a in E. When E is omitted, it is assumed to be R N for some N Z +. K E To be read: K is a compact subset of E. D.W. Stroock, Essentials of Integration Theory f or Analysis, Graduate Texts in Mathematics 262, 235 DOI 10.1007/978-1-4614-1135-2, Springer Science+Business Media, LLC 2011
236 Notation Sets, Functions, and Spaces continued C The mesh size sup{diam(c) : C C} of a cover C 1.1.1 f(x+) & f(x ) The right and left limits of f at x R P(E) The power set of E, consisting of all subsets of E 2.1.2 G(E) &G δ (E) F(E) &F σ (E) C The set of open subsets and the set of countable intersections of open subsets of E The set of closed subsets and the set of countable unions of closed subsets of E The complex numbers N The non-negative integers: N = {0} Z + Q S N 1 Z & Z + C b (E;R) or C b (E;C) e ξ (x) Cc(G;R) or Cc(G;C) C n (G;R) or C n (G;C) L p (µ;r) or L p (µ;c) The set of rational numbers The unit sphere in R N The integers and the positive integers The space of bounded continuous functions from E into R or C The imaginary exponential funtion e i2π(ξ,x) R N The space of continuous, R-valued or C-valued functions having compact support in the open set G The space of n-times continuously differentiable R-valued or C-valued functions The Lebesgue space of R-valued or C-valued functions f for which f p is µ-integrable 6.2.1 l 2 (N;R) The space L 2 (µ;r) when µ is counting measure on N 7.1.2 sgn(x) The signum function: equal to 1 if x 0 and 1 if x < 0 T p (M) The tangent space to M at p 5.2.2 T x Translation by x: T x (y) = x+y 2.2.1 T A The linear transformation determined by the matrix A 2.2.1 JΦ The Jacobian of Φ: JΦ is the absolute value of the determinant of the Jacobian matrix Φ x 5.2.1
Notation 237 Measure Theoretic B E The Borel σ-algebra σ(g(e)) over E B µ The completion of the σ-algebra B with respect to µ 2.1.2 δ a The unit point mass at a H s Hausdorff measure 8.3.4 λ S Lebesgue measure on the set S R N µ ν µ is absolutely continuous with respect to ν Ex. 2.1.27 (R) Γ µ ν µ is singular to ν Ex. 2.1.28 [a,b] Φ µ The pushforward (image) of µ under Φ Ex. 2.1.19 σ(c) The σ-algebra generated by C 2.1.2 f(x)dx ϕ(t) dψ(t) f dµ Γ Equivalent to the Lebesgue integral f dλ Γ RN of f on Γ The Riemann Stieltjes integral of ϕ on [a, b] with respect to ψ The average value the measure µ 1 f dµ of f on Γ with respect to µ(γ) Γ 1.2
absolutely continuous, 39 function, 60 measures, 203 part of a measure, 206 uniformly, 86 additive function, 59 algebra of subsets, 36 almost everywhere, 76 convergence, 76 approximate identity, 169 arc length, 20 of singular, continuous functions, 98 arithmetic geometric mean inequality, 151 axiom of choice, 49 Baire σ-algebra, 219 ball, volume of, 120 Banach space, 146 basis, 179 Bernoulli measure, 54 independence, 55, 61 number, 189 polynomial, 189 Bessel s inequality, 182 Beta function, 133 beta function, 172 Borel σ-algebra, 31 Borel measure, 35 Borel Cantelli Lemma, 38 bounded variation, 12 continuous and discontinuous parts, 19 bump function, 170 A B C Cantor set, 58 Hausdorff dimension of, 233 Carathéodory measurable (C-meaurable), 221 Cauchy Integral Formula, 145 Cauchy Integral Theorem, 143 Cauchy s functional equation, 59 change of variables classical formula, 119 Jacobi s formula for, 122 choice map, 3 closed under set-theoretic operations, 30 co-area formula, 137 complex Hilbert space, 176 concave function, 146 Hessian criterion, 148 contraction, 178 convergence µ-almost everywhere, 76 in µ-measure, 78 convex set, 146 convolution, 165 for the multiplicative group, 173 Young s inequality for, 165 convolution semigroup, 171 Cauchy or Poisson, 172 Weierstrass, 171 coordinate chart, 128 global, 135 countable additivity, 29 countably subadditive, 31 counting measure, 161 cover, 1 exact, 3 more refined, 4 non-overlapping, 1 D Daniell s Theorem, 212 decreasing sequence of sets, 32 diameter, 106 of rectangle, 1 diffeomorphism, 122 Dini s Lemma, 218 direct product, 62 discontinuity, 14 jump, 14 oscillatory, 14 239
240 distribution function of function, 113 of measure, 50 distribution of f in computation of integrals, 115, 161 divergence, 138 Divergence Theorem, 141 dual space, 207 of L p, 208, 220 Euler s product formula, 202 Euler Maclaurin formula, 26 extended real line, 63 extension criterion for measures, 215 Fatou s Lemma, 75, 81 Lieb s version, 77, 81 Lieb s version for L p, 156 finite measure, 31 finitely additive, 208 flow property, 138 Fourier inversion formula for L 1 (λ R N ; C), 193 for L 2 (λ R N ; C), 200 Fourier operator, 198 Fourier series L 1 -theory, 190 L 2 -theory, 186 Fourier transform, 192 for L 1 (λ R N ; C), 192 for L 2 (λ R N ; C), 198 uncertainty principle, 194 Friedrichs mollifier, 169 Fubini s Theorem, 103 function concave, 146 measurable, 64 modulus of continuity of, 11 Fundamental Theorem of Calculus, 10 gamma function, 119 Gauss kernel, 171 gradient, 124 Gram Schmidt orthonormalization procedure, 180 Green s formula, 143 E F G H Hardy s inequality, 173 Hardy Littlewood maximal function, 92, 94 L p -estimate, 162 inequality, 92 Hausdorff dimension, 226 Hausdorff measure, 108, 225 heat flow or Weierstrass semigroup, 171 Helly Bray Theorem, 219 Hermite functions normalized, 197 unnormalized, 194 polynomials, 194 Hermitian inner product, 176 operator, 178 Hessian matrix, 148 Hilbert space orthogonality in, 175 over C, 176 over R, 175 Hölder conjugate, 151, 156 Hölder s inequality, 151 hypersurface, 124 tangent space, 124 idempotent, 178 increasing sequence of sets, 32 independent sets, 38 infinite-dimensional, 179 injective, 122 inner product, 175 Hermitian, 176 integer part, 25 integrable, 70 function the space L 1 (µ; R), 70 uniformly, 86 integration by parts, 9 in R N, 142 isodiametric inequality, 106 Jacobi s Transformation Formula, 122 Jacobian, 121 matrix, 121 Jensen s inequality, 147, 152 I J
241 kernel, 163 L-system, 100 Λ-system, 32 Laplacian, 143 lattice operations, 7 lattice vector integration theory for, 208 Lebesgue measurable, 46 Lebesgue integral exists, 69 notation for, 65 Lebesgue measure, 46 Hausdorff s description, 108 notation for, 117 of a parallelepiped, 58 Lebesgue set, 97 Lebesgue spaces continuity under translation, 166 dual, 220 dual space, 207 L 1 (µ; R), 70 mixed L (p 1,p 2) ((µ 1, µ 2 ); R), 158 Lebesgue s Decomposition Theorem for functions, 95 for measures, 205 Lebesgue s Differentiation Theorem, 96 Lebesgue s Dominated Convergence Theorem, 76, 81 limit inferior of sets, 37 limit of sets, 38 limit superior of sets, 37 linear functional non-negative, 218 tight, 218 linearly independent, 179 lower semicontinuous, 87 lowering operator, 195 K L M Markov s inequality, 68 mean-value property, 144 measurable, 30 function, 64 criteria for, 71 indicator or characteristic, 64 Lebesgue integral of, 68 simple, 64 Lebesgue integral of, 64 Lebesgue, 46 map, 30 µ-measurable set, 34 measurable space, 30 product of, 62 measure, 31 absolutely continuous, 39 Bernoulli, 54 Borel, 35 finite, 31 Hausdorff, 108, 225 image, 37 infinite product of, 217 Lebesgue on R N, 46 non-atomic, 58 probability, 31 product, 103 pullback, 37 pushforward, 37 regular, 35 σ-finite, 82 singular, 39 surface, 132 translation invariant, 47 zero, 76 measure space, 31 complete, 33 completion, 34 mesh size, 3 Minkowski s inequality, 150 continuous version, 159 modulus of continuity, 11 mollification, 169 monotone class, 36 Monotone Convergence Theorem, 75 multi-index, 168 multiplicative group, 172 convolution for, 173 invariant measure for, 173 N non-measurable set, 50 O one-sided, stable law of order 1 2, 171 open δ-hull, 122 orthogonal complement, 179 orthogonal invariance of λ S N 1, 120
242 orthogonal projection operator, 179 orthonormal, 180 outer measure, 220 P parallelepiped, 58 volume of, 59 Parseval s identity, 200 Π-system, 32 Poisson or Cauchy semigroup, 172 Fourier transform, 201 Poisson summation formula, 202 polar coordinates, 118 power set, 30 probability, 31 measure, 31 space, 31 product measure, 103 infinite, 217 pure jump function, 19 R radius, 106 Radon Nikodym derivative, 206 Theorem, 205 raising operator, 195 real Hilbert space, 175 rectangle, 1 volume (vol) of, 1 regular, 35 measure, 35, 37 set, 35 Riemann integral, 3 integrable, 3 integrable with respect to ψ, 8 criterion, 114 of f on J, 3 Riemann sum, 3 lower, 3 upper, 3 vs. Lebesgue, 114 Riemann zeta function, 189 Riemann Lebesgue Lemma, 190, 192 Riemann Stieltjes integral, 8 Riesz Representation Theorem for Hilbert space, 204 for measures, 218 Schwarz s inequality, 152, 175 σ-algebra, 30 Borel, 31 generated by C, 31 σ-finite measure, 82 measure space, 82 signum function (sgn), 17 singular, 39 function, 60 continuous, 73 arc length of, 98 measures, 203 part of a measure, 206 smooth function, 21 smooth region, 139 span, 179 sphere, 106 surface area of, 117, 120 surface measure, 117 square, 45 Steiner symmetrization procedure, 106 Stone s Theorem, 213 Strong Law of Large Numbers, 73 subordination, 172 Sunrise Lemma, 87 surface measure, 132 for graphs, 135 symmetric, 106, 178 S T tangent space, 124 tight family of functions, 86 linear functional, 218 Tonelli s Theorem, 102 translation map, 46 tubular neighborhood, 126 U uncertainty principle, 194, 201 uniform norm, 6 uniformly integrable, 86 uniformly Lipschitz continuous, 57 unitary, 198 upper semicontinuous, 87
243 V variation (Var), 12 negative variation (Var ), 13 positive variation (Var + ), 13 vector field, 137 flow property, 138 vector lattice, 208 integral on, 208 vector semi-lattice, 100 volume of the unit ball, 58 Y Young s inequality, 166 for the multiplicative group, 173