  The purpose of this article is to propose a new method to define and
calculate path integrals over metrics on a K\"ahler manifold. The main idea is
to use finite dimensional spaces of Bergman metrics, as an approximation to the
full space of K\"ahler metrics. We use the theory of large deviations to decide
when a sequence of probability measures on the spaces of Bergman metrics tends
to a limit measure on the space of all K\"ahler metrics. Several examples are
considered.
