INDEX 921

generator, 602growth estimate, 603Hille Yosida theorem, 606strongly continuous, 602

separability of C(H), 769separable metric space

Lindeloff property, 76separated sets, 88separation theorem, 572sequence, 72

Cauchy, 73subsequence, 73

sequential weak* compactness, 558sequentially compact set, 77series

double sum, 64set

F sigma, 253set valued functions

measurability, 271sets, 57

G delta, 253sgn, 39

uniqueness, 41sigma algebra, 237sign of a permutation, 41signed measure, 300

Hahn decomposition, 301signed measures

Hahn decomposition, 300simple functions, 647

approximation, 238singular values, 410slicing measures, 308smooth manifold, 396Sobolev Space

embedding theorem, 391equivalent norms, 391

Sobolev spaces, 391space of continuous martingales, 856

Hilbert space, 879span, 44, 100Sperner’s lemma, 160Steiner symetrization, 450Stirling’s formula, 353stochastic integral

as Stieltjes integral, 891elementary function, 893

linear, 893stochastic process, 786, 807

descriptions, 807Stone Weierstrass theorem, 510Stone’s theorem, 707stong law of large numbers, 792stopped martingale, 862, 864stopped process, 853stopped sub-martingale, 845stopped submartingale, 864stopping time, 795, 831, 835stopping times

conditional expectation, 797strict convexity, 572strictly convex

norm, 147strong law of large numbers, 733, 805strongly measurable, 647

inverse images open sets, 648sub-martingale, 741, 786sub-martingale convergence theorem, 789

continuous case, 850subbasis, 508subspace, 100sums

independent random variables, 729independent variables, 731, 790

supermartingale, 786support of a function, 92, 290, 511symmetric derivative

existence, 342measurable, 342Radon measure, 341upper and lower, 341

symmetric domaindegree, 423

symmetric matrix, 20

tail event, 728Taylor formula, 217Taylor’s formula, 218Taylor’s theorem, 218Tietze extension theorem, 136tight, 766topological space, 501total variation, 348, 622totally bounded, 78, 86totally ordered, 68