Measures, Integrals and Martingales

Описание

Prelude
The purpose of this book is to give a straightforward and yet elementary introduction to measure and integration theory that is within the grasp of second or
third year undergraduates. Indeed, apart from interest in the subject, the only
prerequisites for Chapters 1–13 are a course on rigorous --analysis on the real
line and basic notions of linear algebra and calculus in n. The first few chapters
form a concise (not to say minimalist) introduction to Lebesgue’s approach to
measure and integration, based on a 10-week, 30-hour lecture course for Sussex
University mathematics undergraduates. Chapters 14–24 are more advanced and
contain a selection of results from measure theory, probability theory and analysis. This material can be read linearly but it is also possible to select certain
topics; see the dependence chart on page xi. Although more challenging than the
first part, the prerequisites stay essentially the same and a reader who has worked
through and understood Chapters 1–13 will be well prepared for all that follows.
At some points, one or another concept from point-set topology will be (mostly
superficially) needed; those readers who are not familiar with the topic can look
up the basic results in Appendix B whenever the need arises.
Each chapter is followed by a section of Problems. They are not just drill
exercises but contain variants, excursions from and extensions of the material
presented in the text. The proofs of the core material do not depend on any
of the problems and it is an exception that I refer to a problem in one of the
proofs. Nevertheless I do advise you to attempt as many problems as possible.
The material in the Appendices – on upper and lower limits, basic topology and
the Riemann integral – is primarily intended as back-up, for when you want to
look something up.
Unlike many textbooks this is not an introduction to integration for analysts
or a probabilistic measure theory. I want to reach both (future) analysts and
(future) probabilists, and to provide a foundation which will be useful for both

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