Author | Berberian, Sterling K. author |
---|---|
Title | Fundamentals of Real Analysis [electronic resource] / by Sterling K. Berberian |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1999 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0549-4 |
Descript | XI, 479 p. 98 illus. online resource |
1 Foundations -- ยง1.1. Logic, set notations -- ยง1.2. Relations -- ยง1.3. Functions (mappings) -- ยง1.4. Product sets, axiom of choice -- ยง1.5. Inverse functions -- ยง1.6. Equivalence relations, partitions, quotient sets -- ยง1.7. Order relations -- ยง1.8. Real numbers -- ยง1.9. Finite and infinite sets -- ยง1.10. Countable and uncountable sets -- ยง1.11. Zornโs lemma, the well-ordering theorem -- ยง1.12. Cardinality -- ยง1.13. Cardinal arithmetic, the continuum hypothesis -- ยง1.14. Ordinality -- ยง1.15. Extended real numbers -- ยง1.16. limsup, liminf, convergence in ? -- 2 Lebesgue Measure -- ยง2.1. Lebesgue outer measure on ? -- ยง2.2. Measurable sets -- ยง2.3. Cantor set: an uncountable set of measure zero -- ยง2.4. Borel sets, regularity -- ยง2.5. A nonmeasurable set -- ยง2.6. Abstract measure spaces -- 3 Topology -- ยง3.1. Metric spaces: examples -- ยง3.2. Convergence, closed sets and open sets in metric spaces -- ยง3.3. Topological spaces -- ยง3.4. Continuity -- ยง3.5. Limit of a function -- 4 Lebesgue Integral -- ยง4.1. Measurable functions -- ยง4.2. a.e. -- ยง4.3. Integrable simple functions -- ยง4.4. Integrable functions -- ยง4.5. Monotone convergence theorem, Fatouโs lemma -- ยง4.6. Monotone classes -- ยง4.7. Indefinite integrals -- ยง4.8. Finite signed measures -- 5 Differentiation -- ยง5.1. Bounded variation, absolute continuity -- ยง5.2. Lebesgueโs representation of AC functions -- ยง5.3. limsup, liminf of functions; Dini derivates -- ยง5.4. Criteria for monotonicity -- ยง5.5. Semicontinuity -- ยง5.6. Semicontinuous approximations of integrable functions -- ยง5.7. F. Rieszโs โRising sun lemmaโ -- ยง5.8. Growth estimates of a continuous increasing function -- ยง5.9. Indefinite integrals are a.e. primitives -- ยง5.10. Lebesgueโs โFundamental theorem of calculusโ -- ยง5.11. Measurability of derivates of a monotone function -- ยง5.12. Lebesgue decomposition of a function of bounded variation -- ยง5.13. Lebesgueโs criterion for Riemann-integrability -- 6 Function Spaces -- ยง6.1. Compact metric spaces -- ยง6.2. Uniform convergence, iterated limits theorem -- ยง6.3. Complete metric spaces -- ยง6.4. L1 -- ยง6.5. Real and complex measures -- ยง6.6. L? -- ยง6.7. LP(1 < p < ?) -- ยง6.8.C(X) -- ยง6.9. Stone-Weierstrass approximation theorem -- 7 Product Measure -- ยง7.1. Extension of measures -- ยง7.2. Product measures -- ยง7.3. Iterated integrals, FubiniโTonelli theorem for finite measures -- ยง7.4. FubiniโTonelli theorem for o--finite measures -- 8 The Differential Equation yโ =f (xy) -- ยง8.1. Equicontinuity, Ascoliโs theorem -- ยง8.2. Picardโs existence theorem for yโ =f (xy) -- ยง8.3. Peanoโs existence theorem for yโ =f (xy) -- 9 Topics in Measure and Integration -- ยง9.1. Jordan-Hahn decomposition of a signed measure -- ยง9.2. Radon-Nikodym theorem -- ยง9.3. Lebesgue decomposition of measures -- ยง9.4. Convolution in L1(?) -- ยง9.5. Integral operators (with continuous kernel function) -- Index of Notations