On the weak-hash metric for boundedly finite integer-valued measures

TL;DR

This paper clarifies and corrects foundational proofs regarding the weak-hash metric on boundedly finite integer-valued measures, ensuring the space's completeness and separability are properly established.

Contribution

The authors identify and fix a flaw in the original proofs of key properties of the weak-hash metric, providing corrected arguments for the space's completeness and separability.

Findings

Corrected proofs for the completeness of the measure space

Corrected proofs for the separability of the measure space

Counterexample showing the original assumption does not hold

Abstract

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.