  It is known that the space of boundedly finite integer-valued measures on a
complete separable metric space becomes itself a complete separable metric
space when endowed with the weak-hash metric. It is also known that convergence
under this topology can be characterised in a way that is similar to the weak
convergence of totally finite measures. However, the original proofs of these
two fundamental results assume that a certain term is monotonic, which is not
the case as we give a counterexample. We manage to clarify these original
proofs by addressing specifically the parts that rely on this assumption and
finding alternative arguments.
