  A Banach space has the Schur property when every weakly convergent sequence
converges in norm. We prove a Schur-like property for measures: if a sequence
of finite signed Borel measures on a Polish space is such that it is bounded in
total variation norm and such that for each bounded Lipschitz function the
sequence of integrals of this function with respect to these measures
converges, then the sequence converges in dual bounded Lipschitz norm or
Fortet-Mourier norm to a measure. Moreover, we prove three consequences of this
result: the first is equivalence of concepts of equicontinuity in the theory of
Markov operators, the second is the derivation of weak sequential completeness
of the space of signed Borel measures on Polish spaces from our main result and
the third concerns conditions for the coincidence of weak and norm topologies
on sets of measures that are bounded in total variation norm with additional
properties.
