Counterexamples in Measure and Integration

Erscheinungsjahr 2021
ISBN: 978-1-316-51913-4
Verlag: Cambridge University Press

Buch, Englisch, 330 Seiten, Format (B × H): 198 mm x 249 mm, Gewicht: 862 g

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 (www.cambridge.org/9781316620243).

Autoren/Hrsg.

Schilling, René L

Kühn, Franziska

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

Preface; User's guide; List of topics and phenomena; 1. A panorama of Lebesgue integration; 2. A refresher of topology and ordinal numbers; 3. Riemann is not enough; 4. Families of sets; 5. Set functions and measures; 6. Range and support of a measure; 7. Measurable and non-measurable sets; 8. Measurable maps and functions; 9. Inner and outer measure; 10. Integrable functions; 11. Modes of convergence; 12. Convergence theorems; 13. Continuity and a.e. continuity; 14. Integration and differentiation; 15. Measurability on product spaces; 16. Product measures; 17. Radon–Nikodým and related results; 18. Function spaces; 19. Convergence of measures; References; Index.

Über Autor(innen)

Schilling, René L
René L. Schilling is Professor of Probability Theory at Technische Universität Dresden. His research focuses on stochastic analysis and the theory of stochastic processes.

Kühn, Franziska
Franziska Kühn is Research Assistant at Technische Universität Dresden, where she finished her Ph.D. in 2016. She is interested in the interplay of probability theory and analysis, with a focus on jump processes and non-local operators.