Index Fσ set, 34, 72 Gδ set, 34, 72 Lp norm, 97, 104 σ-algebra, 71 σ-compact, 71 σ-finite, 71, 163 a priori estimate, 176 absolute continuity, 142 absolute integrability, 45, 57, 86 absolutely convergent integral, 86 almost disjoint, 23 almost dominated convergence, 92 almost everywhere, 44 almost everywhere differentiability, 108 almost sure convergence, 96 almost uniform convergence, 95 approximation by an algebra, 79 approximation to the identity, 128, 172 area interpretation of integral, 171 area interpretation of Lebesgue integral, 55 area interpretation of Riemann integral, 14 atomic algebra, 69 axiom of choice, xiv axiom of countable choice, xiv ball, 9 Banach-Tarski paradox, 3 basic jump function, 132 Bernoulli random variables, 199 Besicovitch covering lemma, 125 Bolyai-Gerwien theorem, 11 Boolean algebra, 9, 67 Borel σ-algebra, 72 Borel hierarchy, 73 Borel measure, 126 Borel-Cantelli lemma, 90 bounded variation, 136 bounded variation differentiation theorem, 137 box, 4 bullet-riddled square, 11 Cantor function, 140 Cantor set, 29, 51 Cantor’s theorem, 21 Carath´ eodory measurability, 33, 150 category of measure spaces, 80 change of variables (linear), 34, 55 change of variables (measure), 85 change of variables (series), xii classical derivative, 108 closed dyadic cube, 24 coarsening, 67 compactness, 21 complete measure, 78 completion (measure), 78 continuity of translation, 114 continuous differentiability, 108 continuous probability distribution, 193 continuously differentiable curve, 36 convergence in L1 norm, 95 convergence in L∞ norm, 95 convergence in distribution, 108 203

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