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AuthorBerberian, Sterling K. author
TitleFundamentals of Real Analysis [electronic resource] / by Sterling K. Berberian
ImprintNew York, NY : Springer New York : Imprint: Springer, 1999
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Descript XI, 479 p. 98 illus. online resource


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


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. Zornโ{128}{153}s 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, Fatouโ{128}{153}s lemma -- ยง4.6. Monotone classes -- ยง4.7. Indefinite integrals -- ยง4.8. Finite signed measures -- 5 Differentiation -- ยง5.1. Bounded variation, absolute continuity -- ยง5.2. Lebesgueโ{128}{153}s 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. Rieszโ{128}{153}s โ{128}{156}Rising sun lemmaโ{128}{157} -- ยง5.8. Growth estimates of a continuous increasing function -- ยง5.9. Indefinite integrals are a.e. primitives -- ยง5.10. Lebesgueโ{128}{153}s โ{128}{156}Fundamental theorem of calculusโ{128}{157} -- ยง5.11. Measurability of derivates of a monotone function -- ยง5.12. Lebesgue decomposition of a function of bounded variation -- ยง5.13. Lebesgueโ{128}{153}s 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, Fubiniโ{128}{148}Tonelli theorem for finite measures -- ยง7.4. Fubiniโ{128}{148}Tonelli theorem for o--finite measures -- 8 The Differential Equation yโ{128}{153} =f (xy) -- ยง8.1. Equicontinuity, Ascoliโ{128}{153}s theorem -- ยง8.2. Picardโ{128}{153}s existence theorem for yโ{128}{153} =f (xy) -- ยง8.3. Peanoโ{128}{153}s existence theorem for yโ{128}{153} =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

Mathematics Functions of real variables Mathematics Real Functions


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