INDEX 465

maximum modulus theorem, 379, 383mean value inequality, 79, 120measurability

limit of simple functions, 144measurable, 150

function, 142measurable complex functions

simple functions, 172measurable function

a.e. Borel, 148measurable sets, 151measure, 145

Borel, 155counting, 145finite, 145metric space, 155probability, 145properties, 145

measuresabsolutely continuous, 325decreasing sequences of sets, 145increasing sequences of sets, 145

measures from outer measures, 151Mellen transform, 404meromorphic, 394, 413

on extended complex plane, 415meromorphic functions

a field, 414meromorphic on extended complex

planerational function, 425

metric spacecompact sets, 48complete, 44completely separable, 45open set, 40separable, 45

Minkowski functional, 322Minkowski inequality

integrals, 237Mittag Leffler, 402mixed partial derivatives, 99modular group, 427mollifier, 257monotone convergence theorem, 166Montel’s theorem, 420multi-index, 280

negative part, 169nested interval lemma, 32normal family of functions, 422nowhere differentiable function, 62nowhere differentiable functions, 320numerical range, 450

open ball, 40open set, 40

open cover, 46open mapping theorem, 306, 378, 381open sets, 40

countable basis, 45intersection, 42union, 42

operator norm, 67, 70order of a pole, 394order of a zero, 368ordered

partial, 9totally ordered, 9

orientation, 129, 130simple closed curve, 131

oriented curve, 130orthonormal, 18orthonormal set, 313

maximal, 313outer measure

measurable, 150outer measure on R, 149outer regular, 192

parallelogram identity, 17partial derivatives, 96partial fractions, 24

unique, 26partial order, 9, 313partially ordered set, 9, 202partition of unity, 191periodic functions, 426

Jacobi, 426pi systems, 146Picard

iterates, 227piecewise definition of curves, 126Plancherel theorem, 289, 290point of density, 237polar form complex number, 12