CONTENTS 5

9.11 Completion of a Measure Space . . . . . . . . . . . . . . . . . . . . . . . 2609.12 Vitali Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2629.13 Differentiation of Increasing Functions . . . . . . . . . . . . . . . . . . . 2659.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2689.15 Multifunctions and Their Measurability . . . . . . . . . . . . . . . . . . . 271

9.15.1 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . 2719.15.2 A Special Case When Γ(ω) Compact . . . . . . . . . . . . . . . 2739.15.3 Kuratowski’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 2739.15.4 Measurability of Fixed Points . . . . . . . . . . . . . . . . . . . 2759.15.5 Other Measurability Considerations . . . . . . . . . . . . . . . . 276

9.16 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

10 The Abstract Lebesgue Integral 27910.1 Nonnegative Measurable Functions . . . . . . . . . . . . . . . . . . . . . 279

10.1.1 Riemann Integrals for Decreasing Functions . . . . . . . . . . . 27910.1.2 The Lebesgue Integral for Nonnegative Functions . . . . . . . . 280

10.2 Nonnegative Simple Functions . . . . . . . . . . . . . . . . . . . . . . . 28110.3 The Monotone Convergence Theorem . . . . . . . . . . . . . . . . . . . 28210.4 Other Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28310.5 Fatou’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28310.6 The Integral’s Righteous Algebraic Desires . . . . . . . . . . . . . . . . . 28410.7 The Lebesgue Integral, L1 . . . . . . . . . . . . . . . . . . . . . . . . . . 28410.8 The Dominated Convergence Theorem . . . . . . . . . . . . . . . . . . . 28810.9 Some Important General Theory . . . . . . . . . . . . . . . . . . . . . . 291

10.9.1 Eggoroff’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 29110.9.2 The Vitali Convergence Theorem . . . . . . . . . . . . . . . . . 292

10.10 One Dimensional Lebesgue Stieltjes Integral . . . . . . . . . . . . . . . . 29410.11 The Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . 29710.12 Good Lambda Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 29910.13 Radon Nikodym Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 30010.14 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30410.15 Jensen’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30910.16 Faddeyev’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31010.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

11 Regular Measures 31511.1 Regular Measures in a Metric Space . . . . . . . . . . . . . . . . . . . . 31511.2 Constructing Measures from Functionals . . . . . . . . . . . . . . . . . . 31711.3 The p Dimensional Lebesgue Measure . . . . . . . . . . . . . . . . . . . 32011.4 Maximal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32311.5 Strong Estimates for Maximal Function . . . . . . . . . . . . . . . . . . . 32611.6 The Brouwer Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . 32711.7 Change of Variables, Linear Maps . . . . . . . . . . . . . . . . . . . . . . 32911.8 Differentiable Functions and Measurability . . . . . . . . . . . . . . . . . 33111.9 Change of Variables, Nonlinear Maps . . . . . . . . . . . . . . . . . . . . 33311.10 Mappings Not One to One . . . . . . . . . . . . . . . . . . . . . . . . . . 33711.11 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33911.12 Symmetric Derivative for Radon Measures . . . . . . . . . . . . . . . . . 341