Fundamentals of Real Analysis (Paperback, Softcover Repri)

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.