  In this thesis, the main objects of study are probability measures on the
isomorphism classes of countable, connected rooted graphs. An important class
of such measures is formed by unimodular measures, which satisfy a certain
equation, sometimes referred to as the intrinsic mass transport principle. The
so-called law of a finite graph is an example of a unimodular measure. We say
that a measure is sustained by a countable graph if the set of rooted connected
components of the graph has full measure. We demonstrate several new results
involving sustained unimodular measures, and provide thorough arguments for
known ones. In particular, we give a criterion for unimodularity on connected
graphs, deduce that connected graphs sustain at most one unimodular measure,
and prove that unimodular measures sustained by disconnected graphs are convex
combinations. Furthermore, we discuss weak limits of laws of finite graphs, and
construct counterexamples to seemingly reasonable conjectures.
