  In this article 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 $\nu \in
\mathcal{M}_f(\mathcal{M}_f(X))$, 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. The framework introduced in this article is
motivated by possible applications in biology. It is well suited for modeling
the random evolution of the genealogy of a population in a hierarchical system
with two levels, for example, host-parasite systems or populations which are
divided into colonies. As an example we apply our theory to construct a random
m2m space modeling a two-level coalescent and its genealogy.
