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.