  A pivotal problem in Bayesian nonparametrics is the construction of prior
distributions on the space M(V) of probability measures on a given domain V. In
principle, such distributions on the infinite-dimensional space M(V) can be
constructed from their finite-dimensional marginals---the most prominent
example being the construction of the Dirichlet process from finite-dimensional
Dirichlet distributions. This approach is both intuitive and applicable to the
construction of arbitrary distributions on M(V), but also hamstrung by a number
of technical difficulties. We show how these difficulties can be resolved if
the domain V is a Polish topological space, and give a representation theorem
directly applicable to the construction of any probability distribution on M(V)
whose first moment measure is well-defined. The proof draws on a projective
limit theorem of Bochner, and on properties of set functions on Polish spaces
to establish countable additivity of the resulting random probabilities.
