A representation characterization theorem

Abstract:

Given a field ${\mathbf {F}}$ of subsets of a set $U$ and a real-valued function $T$ defined on a set $S$ of functions from ${\mathbf {F}}$ into $\exp ({\mathbf {R}})$ with bounded range unions, a necessary and sufficient condition is given in order that there be a bounded finitely additive function $\theta$ from ${\mathbf {F}}$ into ${\mathbf {R}}$ such that if $\alpha$ is in $S$, then the integral $\int _U {\alpha (I)\theta (I)}$, as a variational integral, i.e., a refinement-wise limit of appropriate sums over (finite) subdivisions, exists and is $T(\alpha )$.