Measure, Integration & Real Analysis, 2021a

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Contents
2D Lebesgue Measure 47
Additivity of Outer Measure on Borel Sets 47
Lebesgue Measurable Sets 52
Cantor Set and Cantor Function 55
Exercises 2D 60
2E Convergence of Measurable Functions 62
3 Integration 73
Pointwise and Uniform Convergence 62
Egorov’s Theorem 63
Approximation by Simple Functions 65
Luzin’s Theorem 66
Lebesgue Measurable Functions 69
Exercises 2E 71
3A Integration with Respect to a Measure 74
Integration of Nonnegative Functions 74
Monotone Convergence Theorem 77
Integration of Real-Valued Functions 81
Exercises 3A 84
3B Limits of Integrals & Integrals of Limits 88
4 Differentiation 101
Bounded Convergence Theorem 88
Sets of Measure 0 in Integration Theorems 89
Dominated Convergence Theorem 90
Riemann Integrals and Lebesgue Integrals 93
Approximation by Nice Functions 95
Exercises 3B 99
4A Hardy–Littlewood Maximal Function 102
Markov’s Inequality 102
Vitali Covering Lemma 103
Hardy–Littlewood Maximal Inequality 104
Exercises 4A 106
4B Derivatives of Integrals 108
Lebesgue Differentiation Theorem 108
Derivatives 110
Density 112
Exercises 4B 115
Measure, Integration & Real Analysis, by Sheldon Axler