Martingales are a key tool of modern probability theory, in particular, when it comes to a.e. convergence assertions and related limit theorems. The origins of martingale techniques can be traced back to analysis papers by Kac, Marcinkiewicz, Paley, Steinhaus, Wiener and Zygmund from the early 1930s on independent (or orthogonal) functions and the convergence of certain series of functions, see e.g. the paper by Marcinkiewicz and Zygmund which contains many references. The theory of martingales as we know it now goes back to Doob and most of the material of this and the following chapter can be found in his seminal monograph from 1953.

We want to understand martingales as an analysis tool which will be useful for the study of Lp- and almost everywhere convergence and, in particular, for the further development of measure and integration theory. Our presentation differs somewhat from the standard way to introduce martingales – conditional expectations will be defined later in Chapter 22 – but the results and their proofs are pretty much the usual ones. The only difference is that we develop the theory for σ-finite measure spaces rather than just for probability spaces. Those readers who are familiar with martingales and the language of conditional expectations we ask for patience until Chapter 23, in particular Theorem 23.9, when we catch up with these notions.

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 ε-δ analysis on the real line and basic notions of linear algebra and calculus in ℝn. 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.