Abstract. We characterize those positive measures on a Boolean $\sigma $-algebra $A$ which can be represented as the variation of a measure on $A$ with values in an Abelian normed group $G$. We also show that if there exists such a representation, then there is one in which $G$ is an $F^*$-lattice.

AMS Subject Classification (1991): 28B10, 28B05, 28A12, 28A60

Received October 10, 2005, and in revised form July 6, 2006. (Registered under 5936/2009.)