204 Index convergence in mean, 96 convergence in measure, 95 convergence in probability, 96 convolution, 115 countable additivity, 30, 76 countable subadditivity, 18, 77 Cousin’s theorem, 126 create an epsilon of room, 20, 21, 174 cumulative distribution function, 108 Darboux integrability, 13 Darboux integral, 13 de Morgan’s laws, 28 defect version of Fatou’s lemma, 93, 107 density argument, 113 Devil’s staircase, 140 diameter, 20 differentiability, 108 Dini derivative, 129 Dirac measure, 75 directional differentiability, 187 discrete algebra, 68 discrete probability distribution, 193 discretisation, 6 distance (L1), 58 dominated convergence theorem, 91 dominated convergence theorem (sets), 32, 78 domination, 103 dot product, x downward monotone convergence (sets), 32 downwards monotone convergence, 77 Dyadic algebra, 69 dyadic cube, 10 dyadic maximal inequality, 125 dyadic mesh, 24 dyadic nesting property, 25 Egorov’s theorem, 62, 80, 101 elementary algebra, 68 elementary measure, 5 elementary set, 5 escape to horizontal infinity, 63, 87, 97 escape to vertical infinity, 87, 97, 104 escape to width infinity, 87, 97, 104 essential upper bound, 97 essentially uniform convergence, 95 Euclidean space, x event space, 192 existence of non-measurable sets, 37 extended real, xi exterior measure, 149 fast convergence, 101 Fatou’s lemma, 90 finite additivity, 6, 9, 19, 54, 75, 76, 81, 83 finite subadditivity, 9, 18, 76 first fundamental theorem of calculus, 111 first uncountable ordinal, 73 Fubini’s theorem, 170 Fubini-Tonelli theorem, 171 gauge function, 126 generation of algebras, 70, 71 good kernel, 128 gradient, 187 graphs, 9 greedy algorithm, 122 Hahn-Kolmogorov extension, 155 Hahn-Kolmogorov theorem, 154 Hardy-Littlewood maximal inequality, 117, 120, 122, 130 heat kernel, 128 Heaviside function, 160 height (step function), 100 Heine-Borel theorem, 21 Henstock-Kurzweil integral, 147 Hilbert’s third problem, 11 homogeneity, 82 homogeneity (integral), 53, 81 homomorphism, 116 horizontal truncation, 53, 83 inclusion-exclusion principle, 76, 82 indeterminate forms, xi indicator function, x infinite series (absolutely summable), 39 infinite series (unsigned), xii, 39 inner regularity, 32, 196 integration by parts, 140, 161, 190 interval, 4, 156 Jordan algebra, 68 Jordan inner measure, 8 Jordan measurability, 8 Jordan null set, 10 Jordan outer measure, 8, 15 jump function, 132 Kolmogorov extension theorem, 197 Lebesgue algebra, 68

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