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.

Typically, a basic text on functional analysis will only make the briefest of references to general topological vector spaces, before restricting attention to the locally convex case or to Banach spaces. Thus most analysts are aware of the existence of non-locally convex spaces such as Lp (0, 1) for 0 < p < 1 but know very little about them. The neglect of nonlocally convex spaces is easily understood. The basic theory of Banach spaces, which sits at the core of modern functional analysis, may be said to depend on two major principles – the Hahn-Banach theorem and the Closed Graph theorem (which may be taken to include weaker theorems such as the Uniform Boundedness Principle). Working in non-locally convex spaces, even when they are complete and metrizable, requires doing without the Hahn- Banach theorem. The role of the Hahn-Banach theorem may be said to be that of a universal simplifier – infinite-dimensional arguments can be reduced to the scalar case by the use of the ubiquitous linear functional. Thus the problem with non-locally convex spaces is that of “getting off the ground.” This difficulty in even making the simplest initial steps has led some to regard non-locally convex spaces as simply uninteresting. It's our contention, which we hope to justify in these notes, that this attitude is mistaken and that with the aid of fresh techniques one can develop a rich and fulfilling theory.