Suppose that we are given a set X, a Riesz subspace E of Rx, and an increasing linear functional f : E → R. My object in this chapter is to discuss conditions under which f is an ‘integral’, that is, when there is a measure µ on X such that ∫ xdµ exists and is equal to fx for every x ∈ E. A necessary, and nearly sufficient, condition is that f should be ‘sequentially smooth’ [71B–71G]. Further conditions on f and E lead, of course, to stronger results [§§ 72, 73].

The outlines of the theory are not hard to appreciate. Unfortunately, the technical refinements needed for the strongest results are complex, and in their most general forms they are difficult to grasp intuitively. Each extra scrap of information costs us a good deal of hard work. I shall try to summarize the theory in a way which will show its essential structure and maintain reasonable simplicity in the theorems, though the proofs will inevitably be lengthy. The general approach of the first two sections of this chapter is close to that of toposøe T.M.

Sequentially smooth functionals

The first representation theorem I give [71G] is the most natural, in the sense that the hypotheses and consequences are most directly related here. In order not to have to repeat them in the next section, I remove certain parts of the argument into separate lemmas.

The prerequisites for this chapter are few; an acquaintance with linear spaces is the principal requirement. A Riesz space is simply a linear space over the field R of real numbers which has a special kind of partial ordering, and all we need to know about partial orderings will be covered in §§11 and 13. But the theory of Riesz spaces is already rich, and some of the work in §§16 and 17 is far from trivial. It does, however, have to be taken seriously. These are the basic results which will enable us to handle Riesz spaces with assurance and facility.

Partially ordered sets

This section is little more than a list of definitions. As such I suggest that it should be read carefully once, together with the associated examples; but that there is no need to consciously memorize anything. You will find the index perfectly reliable.

Actually the concepts here have applications to every branch of mathematics, and they will mostly be familiar in everything but name. I think it is amusing and instructive to seek such applications out and consciously appreciate them.

In this chapter I shall give versions of those results in elementary measure theory which refer to measure algebras or to L1 and L∞ spaces. The first two sections apply the concepts of §§42–4 to ‘measure rings’, that is, Boolean rings on which a strictly positive countably additive measure is defined. In this case, a true analogy of L1 spaces can be found, and the correspondence between L1 spaces and L# spaces is discussed. All measure rings of any significance are ‘semi-finite’, and consequently their L1 and L# spaces can be identified; this is the basic idea of §52. The next section deals briefly with Dedekind complete measure algebras, which seem to be central to ordinary measure theory. Finally, in §54, the ideas of §45 concerning homomorphisms are reviewed in the new context.

Measure rings

In this section, we shall have only definitions and basic properties. A measure ring is a Boolean ring together with a strictly positive measure; this is an extended-real-valued functional which is additive and sequentially order-continuous on the left. The definition of measure ring which I have chosen [51A] follows the ordinary definition of measure space [61A] in allowing ‘purely infinite’ elements, that is, non-zero elements of infinite measure such that every smaller element is either zero or also of infinite measure.

This book is addressed to functional analysts who would like to understand better the application of their subject to the older discipline of measure theory. The relationship of the two subjects has not always been easy. Measure theory has been the source of many examples for functional analysis; and these examples have been leading cases for some of the most important developments of the general theory. Such a stimulation is, of course, entirely welcome. But there have in addition been several cases in which special results in measure theory have been applied to prove general theorems in analysis. The ordinary functional analyst feels inadequately prepared for these applications, and is exasperated by the intrusion of a large body of knowledge in an unfamiliar style into his own concerns.

My aim therefore is to identify those concepts in measure theory which have most affected functional analysis, and to integrate them into the latter subject in a way consistent with its own structure and habits of thought. The most powerful idea is undoubtedly that of Riesz space, or vector lattice. The principal Banach spaces which measure theory has contributed to functional analysis all have natural partial orderings which render them Riesz spaces. Many of their special properties can be related to the ways in which their orderings, their linear structures and their topologies are connected.

The main purpose of this book is to show how the abstract theory of Riesz spaces may be applied to the study of function spaces arising in measure theory. In Chapters 1–3 I have set out the most important concepts needed for this approach. But in this chapter I propose to open up another line of attack. My eventual aim is to describe the relationship between the measure algebra of a measure space and its function spaces. In order to do this, I demonstrate methods of constructing Riesz spaces from Boolean algebras which, when applied to measure algebras, will produce isomorphic copies of the basic function spaces L1 and L∞. At the same time I shall apply the concepts of the first three chapters to describe the properties of these Riesz spaces.

The technical problems encountered along the way are considerable. An intuitive understanding, however, of the basic constructions S and L∞, is not hard to attain; this is because some relatively easy examples already offer most of the principal aspects of the theory [4XA-4XD]. The construction L# is essentially deeper, and requires faith in Chapters 1–3 to be meaningful at all; its real significance will not appear until § 62.

One of the advantages of this method is that all the constructions are functors, and behave reasonably when the right kind of homomorphism is applied.