Integral representation without additivity

Abstract:

Let $I$ be a norm-continuous functional on the space $B$ of bounded $\Sigma$-measurable real valued functions on a set $S$, where $\Sigma$ is an algebra of subsets of $S$. Define a set function $v$ on $\Sigma$ by: $v (E)$ equals the value of $I$ at the indicator function of $E$. For each $a$ in $B$ let \[ J(a) = \int _{ - \infty }^0 {(v (a \geq \alpha ) - v (S))d\alpha + \int _0^\infty {v (a \geq \alpha )d\alpha .} } \] Then $I = J$ on $B$ if and only if $I(b + c) = I(b) + I(c)$ whenever $(b(s) - b(t))(c(s) - c(t)) \geqslant 0$ for all $s$ and $t$ in $S$.