  This paper studies ergodic properties of certain measures arising in the
dynamics of holomorphic correspondences. These measures, in general, are not
invariant in the classical sense of ergodic theory. We define a notion of
ergodicity, and prove a version of Birkhoff's ergodic theorem in this setting.
In fact, we strengthen this classical result in the setting of rational maps on
the Riemann sphere with the Lyubich measure. We also show the existence of
ergodic measures when a holomorphic correspondence is defined on a compact
complex manifold. Lastly, we give an explicit class of dynamically interesting
measures that are ergodic as in our definition.
