Measures with values in topological groups

This paper is devoted to the study of properties of vector-valued measures defined on a $\sigma$-ring, with values in a topological group. In particular, a necessary and sufficient condition is established in order that the pointwise $B$-boundedness of a sequence of vector-valued measures on the elements of a ring should imply the uniform $B$-boundedness of the sequence on the generated $\sigma$-ring.