Index 205 Lebesgue decomposition, 133 Lebesgue differentiation theorem, 112, 113, 121 Lebesgue exterior measure, 17 Lebesgue inner measure, 33 Lebesgue integral (absolutely integrable), 57 Lebesgue integral (unsigned), 54 Lebesgue measurability, 17 Lebesgue measurability (complex functions), 51 Lebesgue measurability (unsigned functions), 48 Lebesgue outer measure, 16 Lebesgue philosophy, 47 Lebesgue point, 121 Lebesgue-Stieltjes measure, 156 length, x length (intervals), 4 linearity (integral), 13, 14, 44, 46, 58 Lipschitz continuity, 187 Lipschitz differentiation theorem, 138 Littlewood’s first principle, 17, 29, 33, 60 Littlewood’s second principle, 60, 64 Littlewood’s third principle, 60, 62 Littlewood-like principles, 65 local integrability, 121 locally uniform convergence, 61 lower Darboux integral, 13 lower unsigned Lebesgue integral, 52 Lusin’s theorem, 64 Markov’s inequality, 55, 83 mean value theorem, 110 measurability (function), 79 measurability (set), 67 measurable map, 80 measurable morphism, 80 measure, 30, 76 measure space, 76 metric completion, 35 metric entropy, 10 monotone class lemma, 165 monotone convergence theorem, 88, 107 monotone convergence theorem (sets), 32, 77 monotone differentiation theorem, 129 monotonicity (integral), 13, 14, 45, 53, 81, 82 monotonicity (measure), 7, 9, 18, 76 moving bump example, 63 moving bump function, 87 noise tolerance, 47 non-atomic algebra, 69 non-negativity (measure), 9 norm (partition), 12 notation, x null algebra, 68 null set, 26, 78 outer measure, 17, 18, 149, 154 outer regularity, 25 partial derivative, 187 piecewise constant function, 12 piecewise constant integral, 13 pointwise almost everywhere convergence, 61, 95 pointwise convergence, 61, 95 pointwise convergence (sets), 32 Poisson kernel, 128 polytope, 9 pre-measure, 153 probability, 192 probability density, 193 probability measure, 192 probability space, 192 problem of measure, 2 product σ-algebra, 161, 195 product measure, 163, 199 product space, 194 pullback (σ-algebra), 161 pushforward, 85, 196 Rademacher differentiation theorem, 188 Radon measure, 159 recursive description of a σ-algebra, 73 recursive description of Boolean algebra, 70 refinement, 67, 81 reflection, 13, 176 restriction (Boolean algebra), 68 restriction (measure), 83 Riemann integrability, 12 Riemann integral, 12 Riemann sum, 12 Riemann-Stieltjes integral, 160 rising sun inequality, 120 rising sun lemma, 118 Rolle’s theorem, 109 sample space, 192

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