  A Reeb graph is a graphical representation of a scalar function on a
topological space that encodes the topology of the level sets. A Reeb space is
a generalization of the Reeb graph to a multiparameter function. In this paper,
we propose novel constructions of Reeb graphs and Reeb spaces that incorporate
the use of a measure. Specifically, we introduce measure-theoretic Reeb graphs
and Reeb spaces when the domain or the range is modeled as a metric measure
space (i.e.,~a metric space equipped with a measure). Our main goal is to
enhance the robustness of the Reeb graph and Reeb space in representing the
topological features of a scalar field while accounting for the distribution of
the measure. We first introduce a Reeb graph with local smoothing and prove its
stability with respect to the interleaving distance. We then prove the
stability of a Reeb graph of a metric measure space with respect to the
measure, defined using the distance to a measure or the kernel distance to a
measure, respectively.
