This book is a record of a course on functions of a real variable, addressed to first-year graduate students in mathematics, offered in the academic year 1985-86 at the University of Texas at Austin. It consists essentially of the day-by-day lecture notes that I prepared for the course, padded up with the exercises that I seemed never to have the time to prepare in advance; the structure and contents of the course are preserved faithfully, with minor cosmetic changes here and there.
PrefaceCHAPTER 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 1.1 1. 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 inCHAPTER 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 spacesCHAPTER 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 functionCHAPTER 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 measuresCHAPTER. 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-integrabilityCHAPTER 6 Function Spaces 6.1. Compact metric spaces 6.2. Uniform convergence, iterated limits theorem 6.3. Complete metric spaces 6.4. LI 6.5. Real and complex measures 6.6. Loo 6.7. Lp (1
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