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.