On the existence and representation of integrals (Q2537996)

The author's introduction reads as follows. ``Suppose that \(\Omega\) is set, \(R\) is a non-empty collection of subsets of \(\Omega\), and \(D\) is the collection of finite non-empty subsets of \(R\) to which \(M\) belongs only in case \(M^*\), the union of all the members of \(M\), is in \(R\) and the members of \(M\) are relatively prime in \(R\), i.e., if \(A\) and \(B\) are in \(M\) then there is no non-empty member of \(R\) which is contained in both \(A\) and \(B\). We will assume that each non-empty \(A\) in \(R\) contains a point \(x\) such that if \(M\) is in \(D\) and \(A\) is in \(M\) then no other member of \(M\) contains \(x\). Let \(B(\Omega, R)\) denote the closure in the space of functions from \(\Omega\) to the number-plane which have bounded final sets of the linear space spanned by the characteristic functions of members of \(R\) with respect to the supremum norm \(|\,\cdot\,|\). We will assume that \(B(\Omega, R)\) is an algebra. An integral on \(B(\Omega, R)\times R\) is a function \(K\) from \(B(\Omega, R)\times R\) to the number-plane such that (1) for each \((f, A)\) in \(B(\Omega, R)\times R\) \(K[\;,A]\) is a linear functional on \(B(\Omega, R)\) and \(K[f,\;]\) is additive on \(R\), i.e., \(K(f, M^*)=\sum_{H\text{ in }M} K(f, H)\) for each \(M\) in \(D\), and (2) there is an additive function \(\lambda\) from \(R\) to the non-negative numbers such that \(| K(f, A)| \leq | 1_A f| \lambda(A)\), for each \((f, A)\) in \(B(\Omega, R)\times R\). This paper is concerned with the existence and representation of integrals on \(B(\Omega, R)\times R\).'' Much of the paper is written in the style of the initial sentences of these two paragraphs and is therefore incomprehensible to the reviewer. One of the results is asserted to be an extension of a representation theorem of \textit{J. S. MacNerney [Bull. Am. Math. Soc. 69, 314--329 (1965; Zbl 0113.09003)]; that paper is the only item In the bibliography.}