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/dv of one measure with respect to another for suitable measures μ, v: but the point function dμ/dv 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/dv 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 R of 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. In §9.4 we describe the Daniell integral and show that, under suitable conditions this can be obtained in terms of a measure.

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 ℳ as a representative of the class of ℱ-measurable functions which are equal to f a.e. As is usual we will use the same notation f or 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.

There are many ways of developing the theory of measure and integration. In the present book measure is studied first as the primary concept and the integral is obtained later by extending its definition from the special case of ‘simple’ functions using monotone limits. The theory is presented for general measure spaces though at each stage Lebesgue measure and the Lebesgue integral in Rn are considered as the most important example, and the detailed properties are established for the Lebesgue case.

The book is designed for use either in the final undergraduate year at British universities or as a basic text in measure theory at the postgraduate level. Though the subject is developed as a branch of pure mathematics, it is presented in such a way that it has immediate application to any branch of applied mathematics which requires the basic theory of measure and integration as a foundation for its mathematical apparatus. In particular, our development of the subject is a suitable basis for modern probability theory – in fact this book first appeared as the initial section of the book Introduction to measure and probability (Cambridge University Press, 1966) written jointly with J. F. C. Kingman.

The book is largely self-contained. The first two chapters contain the essential parts of set theory and point set topology; these could well be omitted by a reader already familiar with these subjects. Chapters 3 and 4 develop the theory of measure by the usual process of extension from ‘simple sets’ to those of a larger class, and the properties of Lebesgue measure are obtained.

Sets

We do not want to become involved in the logical foundations of mathematics. In order to avoid these we will adopt a rather naive 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. If E ⊂ F and there is at least one element in F but not in E, we say that E is a proper subset of F.

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. When we speak of a metric space X we mean the set X together with a particular ρ satisfying conditions (i), (ii) and (iii) above.