  We introduce a family of Banach spaces of measures, each containing the set
of measures with density of bounded variation. These spaces are suitable for
the study of weighted transfer operators of piecewise-smooth maps of the
interval where the weighting used in the transfer operator is not better than
piecewise H\"older continuous and the partition on which the map is continuous
may possess a countable number of elements. For such weighted transfer
operators we give upper bounds for both the spectral radius and for the
essential spectral radius.
