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

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}.