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.