Because a point in space can be represented by a triple of real numbers, all geometric properties of spatial figures can be expressed in terms of real numbers. Hence one can theoretically understand geometry solely through analysis. But a true appreciation of geometry requires not only analytical technique but also intuition of geometric objects. The same holds for probability theory. The modern theory of probability is formulated in terms of measures and integrals and so is part of modern analysis from the logical viewpoint. But to really enjoy probability theory, one should grasp the orientation of development of the theory with intuitive insight into random phenomena. The purpose of this book is to explain basic probabilistic concepts rigorously as well as intuitively.

In Chapter 1 we restrict ourselves to trials with a finite number of outcomes. The concepts discussed here are those of elementary probability theory but are dealt with from the advanced standpoint. We hope that the reader appreciates how random phenomena are discussed mathematically without being annoyed with measure-theoretic complications.

In the subsequent chapters we expect the reader to be more or less familiar with basic facts in measure theory.

In Chapter 2 we discuss the properties of those probability measures that appear in this book.

In Chapter 3 we explain the fundamental concepts in probability theory such as events, random variables, independence, conditioning, and so on. We formulate these concepts on a perfect separable complete probability space. The additional conditions “perfectness” and “separability” are imposed to construct the theory in a more natural way. The reader will see that such conditions are satisfied in all problems appearing in applications.

In the standard textbook conditional probability is defined with respect to a-algebras of subsets of the sample space (Doob's definition). Here we first define it with respect to decompositions of the sample space (Kolmogorov's definition) to make it easier for the reader to understand its intuitive meaning and then explain Doob's definition and the relation between these two definitions.

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.