Convergence of metric two-level measure spaces

摘要

We extend the notion of metric measure spaces to so-called metric two-level measure spaces (m2m spaces): An m2m space (X, r, nu) is a Polish metric space (X, r) equipped with a two-level measure nu is an element of M-f (M-f (X)), i.e. a finite measure on the set of finite measures on X. We introduce a topology on the set of (equivalence classes of) m2m spaces induced by certain test functions (i. e. the initial topology with respect to these test functions) and show that this topology is Polish by providing a complete metric.