This volume consists of proofs of the problems in the monograph Real Analysis: Theory of Measure and Integration, 3rd Edition. Alternate proofs are included when appropriate to show different approaches to the problem or different techniques in constructing a proof. We hope that this volume will be helpful to those who read Real Analysis in self-study and have no easy access to help.

Preface viii

§1 Measure on a σ-algebra of Sets 1

§2 Outer Measures 43

§3 Lebesgue Measure on R 46

§4 Measurable Functions 78

§5 Completion of Measure Space 104

§6 Convergence a.e. and Convergence in Measure 107

§7 Integration of Bounded Functions on Sets of Finite Measure 132

§8 Integration of Nonnegative Functions 160

§9 Integration of Measurable Functions 192

§10 Signed Measures 266

§11 Absolute Continuity of a Measure 277

§12 Monotone Functions and Functions of Bounded Variation 290

§13 Absolutely Continuous Functions 314

§16 The LP Spaces 353

§17 Relation among the LP Spaces 386

§18 Bounded Linear Functionals on the LP Spaces 424

§22 Lebesgue-Stieltjes Measure Spaces 431

§23 Product Measure Spaces 443

§24 Lebesgue Measure Space on the Euclidean Space 466

§25 Differentiation on the Euclidean Space 473

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