  The dual space of the C*-algebra of bounded uniformly continuous functions on
a uniform space carries several natural topologies. One of these is the
topology of uniform convergence on bounded uniformly equicontinuous sets, or
the UEB topology for short. In the particular case of a topological group and
its right uniformity, the UEB topology plays a significant role in the
continuity of convolution. In this paper we derive a useful characterisation of
bounded uniformly equicontinuous sets on locally compact groups. Then we
demonstrate that for every locally compact group G the UEB topology on the
space of finite Radon measures on G coincides with the right multiplier
topology. In this sense the UEB topology is a generalisation to arbitrary
topological groups of the multiplier topology for locally compact groups. In
the final section we prove results about UEB continuity of convolution.
