Multi-occupation field generates the Borel-sigma-field of loops

TL;DR

This paper proves that the Borel-sigma-field on the space of cadlag loops in a Polish space is generated by multi-occupation field functionals, extending discrete case results to continuous settings.

Contribution

It establishes that the multi-occupation field generates the Borel-sigma-field for loop spaces in Polish spaces, generalizing previous discrete case findings.

Findings

The loop space is Polish under the Skorokhod metric.

The Borel-sigma-field is generated by multi-occupation field functionals.

The result extends discrete loop space results to continuous Polish spaces.

Abstract

In this article, we consider the space of c\`{a}dl\`{a}g loops on a Polish space $S$. The loop space can be equipped with a ``Skorokhod'' metric. Moreover, it is Polish under this metric. Our main result is to prove that the Borel-$\sigma$-field on the space of loops is generated by a class of loop functionals: the multi-occupation field. This result generalizes the result in the discrete case, see \cite{loop}.