Fundamentals of Real Analysis: (Universitext)

Integration theory and general topology form the core of this textbook for a first-year graduate course in real analysis. After the foundational material in the first chapter (construction of the reals, cardinal and ordinal numbers, Zorn's lemma and transfinite induction), measure, integral and topology are introduced and developed as recurrent themes of increasing depth. The treatment of integration theory is quite complete (including the convergence theorems, product measure, absolute continuity, the Radon-Nikodym theorem, and Lebesgue's theory of differentiation and primitive functions), while topology, predominantly metric, plays a supporting role. In the later chapters, integral and topology coalesce in topics such as function spaces, the Riesz representation theorem, existence theorems for an ordinary differential equation, and integral operators with continuous kernel function. In particular, the material on function spaces lays a firm foundation for the study of functional analysis.

Table of Contents:
1 Foundations.- §1.1. Logic, set notations.- §1.2. Relations.- §1.3. Functions (mappings).- §1.4. Product sets, axiom of choice.- §1.5. Inverse functions.- §1.6. Equivalence relations, partitions, quotient sets.- §1.7. Order relations.- §1.8. Real numbers.- §1.9. Finite and infinite sets.- §1.10. Countable and uncountable sets.- §1.11. Zorn’s lemma, the well-ordering theorem.- §1.12. Cardinality.- §1.13. Cardinal arithmetic, the continuum hypothesis.- §1.14. Ordinality.- §1.15. Extended real numbers.- §1.16. limsup, liminf, convergence in ?.- 2 Lebesgue Measure.- §2.1. Lebesgue outer measure on ?.- §2.2. Measurable sets.- §2.3. Cantor set: an uncountable set of measure zero.- §2.4. Borel sets, regularity.- §2.5. A nonmeasurable set.- §2.6. Abstract measure spaces.- 3 Topology.- §3.1. Metric spaces: examples.- §3.2. Convergence, closed sets and open sets in metric spaces.- §3.3. Topological spaces.- §3.4. Continuity.- §3.5. Limit of a function.- 4 Lebesgue Integral.- §4.1. Measurable functions.- §4.2. a.e..- §4.3. Integrable simple functions.- §4.4. Integrable functions.- §4.5. Monotone convergence theorem, Fatou’s lemma.- §4.6. Monotone classes.- §4.7. Indefinite integrals.- §4.8. Finite signed measures.- 5 Differentiation.- §5.1. Bounded variation, absolute continuity.- §5.2. Lebesgue’s representation of AC functions.- §5.3. limsup, liminf of functions; Dini derivates.- §5.4. Criteria for monotonicity.- §5.5. Semicontinuity.- §5.6. Semicontinuous approximations of integrable functions.- §5.7. F. Riesz’s “Rising sun lemma”.- §5.8. Growth estimates of a continuous increasing function.- §5.9. Indefinite integrals are a.e. primitives.- §5.10. Lebesgue’s “Fundamental theorem of calculus”.- §5.11. Measurability of derivates of a monotone function.- §5.12. Lebesgue decomposition of a function of bounded variation.- §5.13. Lebesgue’s criterion for Riemann-integrability.- 6 Function Spaces.- §6.1. Compact metric spaces.- §6.2. Uniform convergence, iterated limits theorem.-§6.3. Complete metric spaces.- §6.4. L1.- §6.5. Real and complex measures.- §6.6. L?.- §6.7. LP(1 < p < ?).- §6.8.C(X).- §6.9. Stone-Weierstrass approximation theorem.- 7 Product Measure.- §7.1. Extension of measures.- §7.2. Product measures.- §7.3. Iterated integrals, Fubini—Tonelli theorem for finite measures.- §7.4. Fubini—Tonelli theorem for o--finite measures.- 8 The Differential Equation y’ =f (xy).- §8.1. Equicontinuity, Ascoli’s theorem.- §8.2. Picard’s existence theorem for y’ =f (xy).- §8.3. Peano’s existence theorem for y’ =f (xy).- 9 Topics in Measure and Integration.- §9.1. Jordan-Hahn decomposition of a signed measure.- §9.2. Radon-Nikodym theorem.- §9.3. Lebesgue decomposition of measures.- §9.4. Convolution in L1(?).- §9.5. Integral operators (with continuous kernel function).- Index of Notations.

Review :
"This book is very well organized and clearly written and contains an adequate supply of exercises. If one is comfortable with the choice of topics in the book, it would be a good candidate for a text in a graduate real analysis course." -- MATHEMATICAL REVIEWS