Fundamentals of Real Analysis

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.

"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