# 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.

## Key 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.