206 Index second fundamental theorem of calculus, 110, 138, 139, 143, 145 seminorm, 57 simple function, 41 simple integral, 42, 81 Solovay’s theorem, 36 space-filling curve, 36 Steinhaus theorem, 115, 127 step function, 60 strong derivative, 108 sub-null set, 78 subadditivity (integral), 53 sums of measures, 77 superadditivity, 82 superadditivity (integral), 53 support, 44 symmetric difference, 5 tagged partition, 12 tail support, 100 Tonelli’s theorem, 166–168 Tonelli’s theorem (series), xiii, xiv Tonelli’s theorem (sums and integrals), 89 total differentiability, 187 total variation, 136 translation (of a set in Euclidean space), 5 translation invariance, 7, 9, 34, 55 triangle inequality, 59 trivial algebra, 68 typewriter sequence, 97 uniform continuity, 142 uniform convergence, 61, 95 uniform integrability, 104 uniformly almost everywhere convergence, 95 uniqueness of antiderivative, 110 uniqueness of Jordan measure, 11 uniqueness of Lebesgue measure, 35 uniqueness of the Lebesgue integral, 55 uniqueness of the Riemann integral, 14 uniqueness of the unsigned integral, 93 unsigned integral, 82 upper Darboux integral, 13 upper unsigned Lebesgue integral, 53 upward monotone convergence (sets), 31 upwards monotone convergence, 77 vertical truncation, 53, 83 Vitali-type covering lemma, 122 volume (box), 5 weak derivative, 189 Weierstrass function, 129 width (step function), 100 zero measure, 75

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