  A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r)
is a complete and separable metric space and mu is a probability measure on XxI
for some Polish space I of possible marks. We study the space of all
(equivalence classes of) marked metric measure spaces for some fixed I. It
arises as state space in the construction of Markov processes which take values
in random graphs, e.g. tree-valued dynamics describing randomly evolving
genealogical structures in population models. We derive here the topological
properties of the space of mmm-spaces needed to study convergence in
distribution of random mmm-spaces. Extending the notion of the Gromov-weak
topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the
marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish
space. We give a characterization of tightness for families of distributions of
random mmm- spaces and identify a convergence determining algebra of functions,
called polynomials.
