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Author Schilling, René L., author

Title Counterexamples in measure and integration / René L. Schilling, Franziska Kühn
Published Cambridge ; New York, NY: Cambridge University Press, 2021

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Description 1 online resource
Contents 1.9 Extension of Set Functions and Measures -- 1.10 Signed Measures and Radon-Nikodým -- 1.11 A Historical Aperçu From the Beginnings Until 1854 -- 1.12 Appendix: H. Lebesgue's Seminal Paper -- 2 A Refresher of Topology and Ordinal Numbers -- 2.1 A Modicum of Point-Set Topology -- 2.2 The Axiom of Choice and Its Relatives -- 2.3 Cardinal and Ordinal Numbers -- 2.4 The Ordinal Space -- 2.5 The Cantor Set: A Nowhere Dense, Perfect Set -- 2.6 The Cantor Function and Its Inverse -- 3 Riemann Is Not Enough -- 3.1 The Riemann-Darboux upper integral is not additive
3.2 Why one should define Riemann integrals on bounded intervals -- 3.3 There are no unbounded Riemann integrable functions -- 3.4 A function which is not Riemann integrable -- 3.5 Yet another function which is not Riemann integrable -- 3.6 A non-Riemann integrable function where a sequence of Riemann sums is convergent -- 3.7 A Riemann integrable function without a primitive -- 3.8 A Riemann integrable function whose discontinuity points are dense -- 3.9 Semicontinuity does not imply Riemann integrability -- 3.10 A function which has the intermediate value property but is not Riemann integrable
3.16 The space of Riemann integrable functions is not complete -- 3.17 An example where integration by substitution goes wrong -- 3.18 A Riemann integrable function which is not Borel measurable -- 3.19 A non-Riemann integrable function f which coincides a.e. with a continuous function -- 3.20 A Riemann integrable function on R[sup(2)] whose iterated integrals arenot Riemann integrable -- 3.21 Upper and lower integrals do not work for the Riemann-Stieltjes integral -- 3.22 The Riemann-Stieltjes integral does not exist if integrand and integrator have a common discontinuity -- 4 Families of Sets
Summary "Often it is more instructive to know 'what can go wrong' and to understand 'why a result fails' than to plod through yet another piece of theory. In this text, the authors gather more than 300 counterexamples - some of them both surprising and amusing - showing the limitations, hidden traps and pitfalls of measure and integration. Many examples are put into context, explaining relevant parts of the theory, and pointing out further reading. The text starts with a self-contained, non-technical overview on the fundamentals of measure and integration. A companion to the successful undergraduate textbook Measures, Integrals and Martingales, it is accessible to advanced undergraduate students, requiring only modest prerequisites. More specialized concepts are summarized at the beginning of each chapter, allowing for self-study as well as supplementary reading for any course covering measures and integrals. For researchers, it provides ample examples and warnings as to the limitations of general measure theory. This book forms a sister volume to René Schilling's other book Measures, Integrals and Martingales"-- Provided by publisher
Bibliography Includes bibliographical references and index
Notes Description based on print version record and CIP data provided by publisher; resource not viewed
Subject Measure theory.
Integrals.
MATHEMATICS / Mathematical Analysis.
Integrales
Teoría de la medida
Integrals
Measure theory
Genre/Form Electronic books
Form Electronic book
Author Kühn, Franziska, 1989- author.
LC no. 2021017091
ISBN 9781009001625
1009001620
9781009003797
1009003798