Metric space

In the first chapter we were concerned with abstract sets where no structure in the set was assumed or used. In practice, most useful spaces do have a structure which can be described in terms of a class of subsets called ‘open’. By far the most convenient method of obtaining this class of open sets is to quantify the notion of nearness for each pair of points in the space. A non-empty set X together with a ‘distance’ function ρ:X × X → R is said to form a metric space provided that

1. (i) ρ(y,x) = ρ(x,y) ≥ 0 for all x,y ∈ X;

2. (ii) ρ(x, y) = 0 if and only if x = y;

3. (iii) ρ(x, y) ≤ ρ(x, z) + ρ(y, z) for all x, y, z ∈ X.

The real number ρ(x, y) should be thought of as the distance from x to y. Note that it is possible to deduce conditions (i), (ii) and (iii) from a smaller set of axioms: this has little point as all the conditions agree with the intuitive notion of distance. Condition (iii) for ρ is often called the triangle inequality because it says that the lengths of two sides of a triangle sum to at least that of the third. Condition (ii) ensures that ρ distinguishes distinct points of X, and (i) says that the distance from y to x is the same as the distance from x to y.

In the present book most of the theory of measure and integration has been developed in abstract spaces, and we have used the properties of special spaces only to illustrate the general theory. The present chapter, apart from §9.4, is devoted to a discussion of properties which depend essentially on the structure of the space.

The first question considered is that of point-wise differentiation. In the Radon–Nikodym theorem 6.7 we defined the derivative dμ/dν of one measure with respect to another for suitable measures μ, ν: but the point function dμ/dν obtained is only determined in the sense that the equivalence class of functions equal almost everywhere is uniquely defined. This means that at no single point (except for those points which form sets of positive measure) is the derivative defined by the Radon–Nikodym theorem. In order to define dμ/dν at a point x, the local topological structure of the space near x has to be considered. It is possible to develop this local differentiation theory in fairly general spaces, but only at the cost of complicated and rather unnatural additional conditions: we have decided instead to give the detailed theory only in the space Rof real numbers where the term derivative has a clear elementary meaning.

There are several ways of defining an integral with properties similar to those obtained in Chapter 5. So far in this book we have considered definitions which start from a given measure defined on a suitable class of sets.

Throughout this chapter we will assume (unless stated otherwise) that (Ω,F,μ) is a σ-finite measure space, and that the σ-field F is complete with respect to μ. This implies that if f:Ω→R*, g: Ω→R* are functions such that f is F-measurable and f = g a.e., then g is also F-measurable. Thus, if M is the class of functions f: Ω → R* which are F-measurable, we say that f1,f2 in M are equivalent if f1 = f2 a.e. This clearly defines an equivalence relation in M and we can form the space M of equivalence classes with respect to this relation. When we think of a function f of M as an element of M we are really thinking of f as a representative of the class of F-measurable functions which are equal to fa.e. As is usual we will use the same notation f for an element of M and M. We can think of M or M as an abstract space, and the definition of convergence if given in terms of a metric will then impose a topological structure on the space. We will consider several such notions of convergence of which some, but not all, can be expressed in terms of a metric in M. We will obtain the relationships between different notions of convergence, and in each case prove that the space is complete in the sense that for any Cauchy sequence there is a limit function to which the sequence converges.

Sets

We do not want to become involved in the logical foundations of mathematics. In order to avoid these we will adopt a rather naïve attitude to set theory. This will not lead us into difficulties because in any given situation we will be considering sets which are all contained in (are subsets of) a fixed set or space or suitable collections of such sets. The logical difficulties which can arise in set theory only appear when one considers sets which are ‘too big’–like the set of all sets, for instance. We assume the basic algebraic properties of the positive integers, the real numbers, and Euclidean spaces and make no attempt to obtain these from more primitive set theoretic notions. However, we will give an outline development (in Chapter 2) of the topological properties of these sets.

In a space X a set E is well defined if there is a rule which determines, for each element (or point) x in X, whether or not it is in E. We write x∈E (read ‘x belongs to E’) whenever x is an element of E, and the negation of this statement is written x ∉ E. Given two sets E, F we say that E is contained in F, or E is a subset of F, or F contains E and write E ⊂ F if every element x in E also belongs to F.

What is an integral?

Historically the concept of integration was first considered for real functions of a real variable where either the notion of ‘the process inverse to differentiation’ or the notion of ‘area under a curve’ was the starting point. In the first case a real number was obtained as the difference of two values of the ‘indefinite’ integral, while the second case corresponds immediately to the ‘definite’ integral. The so-called ‘fundamental theorem of the integral calculus’ provided the link between the two ideas. Our discussion of the operation of integration will start from the notion of a definite integral, though in the first instance the ‘interval’ over which the function is integrated will be the whole space. Thus, for ‘suitable’ functions f: Ω → R* we want to define the integral I(f) as a real number. The ‘suitable’ functions will be called integrable and I(f) will be called the integral of f.

Before defining such an operator I, we examine the sort of properties I should have before we would be justified in calling it an ‘integral’. Suppose then that A is a class of functions f: Ω → R*, and I:A → R defines a real number for every f∈A. Then we want I to satisfy:

1. (i) f∈A, f(x) ≥ 0 all x∈Ω ⇒ I(f) ≥ 0, that is I preserves positivity;

2. […]

We decided to write this book largely as a result of experience in teaching at the Instructional Conference on Probability held at Durham in 1963 under the auspices of the London Mathematical Society. It seemed that a proper treatment of probability theory required an adequate background in the theory of finite measures in general spaces. The first part of the book attempts to set out this material in a form which not only provides an introduction for the intending specialist in measure theory, but also meets the needs of students of probability.

The theory of measure and integration is presented in the first instance for general spaces, though at each stage Lebesgue measure and the Lebesgue integral are considered as important examples, and their detailed properties are obtained. An introduction to functional analysis is given in Chapters 7 and 8; this contains not only the material (such as the various notions of convergence) which is relevant to probability theory, but also covers the basic theory of L2-spaces important, for instance, in modern physics.

The second part of the book is an account of the fundamental theoretical ideas which underlie the applications of probability in statistics and elsewhere. The treatment leans heavily on the machinery developed in the first half of the book, and indeed some of the most important results are merely restatements of standard theorems of measure theory.