Gentlemen,

Observing your enthusiastic desire for me to make a detailed, examination of the writings of M. Descartes, I felt myself obliged, in this matter, to go along with the wishes of such very good friends. In complying with your request I hope to make you realize the great regard which I have for you, and also to establish the inadequacy of my own intellectual powers, so that in future you may show me a little more indulgence, if I require it, and not press me so hard if my performance here turns out to be inadequate.

M. Descartes is in my judgement a man of the highest intellect and the utmost modesty – a man such as even Momus, were he now with us, would approve of. ‘I am thinking’, he says, ‘therefore I exist; indeed, I am thought itself – I am a mind.’ Granted. ‘But in virtue of thinking, I possess within me ideas of things, and in particular an idea of a supremely perfect and infinite being.’ True again. ‘However I am not the cause of this idea, since I do not measure up to its objective reality; hence something more perfect than myself is its cause, and accordingly there exists something besides myself, something more perfect than I am. This is someone who is not a being in any ordinary sense but who simply and without qualification embraces the whole of being within himself, and is as it were the ultimate original cause, as Dionysius says in chapter eight of the Divina Nomina.’

There are many matters which remain to be investigated concerning the attributes of God and the nature of myself, or my mind; and perhaps I shall take these up at another time. But now that I have seen what to do and what to avoid in order to reach the truth, the most pressing task seems to be to try to escape from the doubts into which I fell a few days ago, and see whether any certainty can be achieved regarding material objects.

But before I inquire whether any such things exist outside me, I must consider the ideas of these things, in so far as they exist in my thought, and see which of them are distinct, and which confused.

Quantity, for example, or ‘continuous’ quantity as the philosophers commonly call it, is something I distinctly imagine. That is, I distinctly imagine the extension of the quantity (or rather of the thing which is quantified) in length, breadth and depth. I also enumerate various parts of the thing, and to these parts I assign various sizes, shapes, positions and local motions; and to the motions I assign various durations.

Not only are all these things very well known and transparent to me when regarded in this general way, but in addition there are countless particular features regarding shape, number, motion and so on, which I perceive when I give them my attention.

Some years ago I regularly gave a traditional course on metric spaces to second-year special honours mathematics students. I was then asked to give a watered-down version of the same material to a class of combined honours students (who were doing several subjects, including mathematics, at a more general level) but, to put it mildly, the course was not a success. It was impossible to motivate students to generalise real analysis when they had never understood it in the first place and certainly could not remember much of it. It was also counter-productive to start the course by revising real analysis because that convinced the students that this was ‘just another analysis course’ and their interest was lost for evermore.

So when I gave the course again the following year I decided to turn the material inside out and to start with the applications (namely the use of contractions in solving a wide range of equations). This meant that the first chapter was a revision of some iterative techniques used to obtain approximations to solutions of equations. This immediately captured the interest of the class: they enjoyed using their calculators and writing programs to solve the equations. Some of the ideas were entirely new to them; for example using iteration to solve an equation with constraints, or solving a differential equation by iterating with an integral and obtaining a sequence of functions.

The second and third chapters were more traditional but the big difference was that the need for distance, function space, closed set, and so on, had been anticipated and motivated. Another difference was that, having approached the subject via iteration, it was then natural to define all the concepts in terms of sequences: hence closed sets (rather than open ones) formed the basis of the approach.

For most students the fourth chapter was the highlight of the course. It consisted of the contraction mapping principle and the use of its algorithmic proof in solving equations.

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.