Adapted sequences of integrable functions arise naturally in probability theory. Martingales, submartingales and supermartingales especially are very important to probabilists since they serve as mathematical models for many probabilistic phenomena. Consider for instance the fortune of a gambler. The martingale condition corresponds to the situation where this fortune remains constant in the sense of conditional mean. The supermartingale condition corresponds to the situation where at each play the game is unfavorable to the gambler in the same sense, while the submartingale condition corresponds to the situation where at each play the game is favorable in that sense. It is therefore clear that these notions are extremely important in probability theory, and so they have been heavily studied. One of the most interesting questions is when (and to what) does such an adapted sequence converge almost everywhere?

Such classes of adapted sequences do not only have interest in probability theory. They have also been used in other branches of mathematics such as potential theory, dynamical systems and many others.

However it is my feeling that not many analysts are used to dealing with martingales. That is even more the case with extensions of the martingale notion, involving stopping times. Nevertheless stopping time techniques do have many applications in real or functional analysis. This is what this book is about : to be of use to probabilists (of course) but also to analysts, by introducing them to the most important stopping time techniques.

Martingale theory is one of the most powerful tools of the modern probabilist. Its intuitive appeal and intrinsic simplicity combine with an impressive array of stability properties which enables us to construct and analyse many concrete examples within an abstract mathematical framework. This makes martingales particularly attractive to the student with a good background in pure mathematics wishing to find a convenient route into modern probability theory. The range of applications is enhanced by the construction of stochastic integrals and a martingale calculus.

This text has grown out of graduate lecture courses given at the University of Hull to students with a strong background in analysis but with little previous exposure to stochastic processes. It represents an attempt to make the ‘general theory of processes’ and its application to the construction of stochastic integrals accessible to such readers. As may be expected, the material is drawn largely from the work of Meyer and Dellacherie, but the influence of such authors as Elliott, Kussmaul, Neveu and Kallianpur will also be evident. I have not attempted to give credit for particular results: most of the material covered can now be described as standard, and I make no claims of originality. The Appendix by Chris Barnett and Ivan Wilde contains recent work on non-commutative integrals, some of which is presented here for the first time.

In general I have tried to follow the simplest, thus not always the shortest, route to the principal results, often pausing for motivation through familiar concepts.