Probability Measures on Product Spaces with Uniform Metrics

Abstract

For a countable product of complete separable metric spaces, with a topology induced by a uniform metric, the σ-algebra generated by the open balls, which was introduced by Dudley (1966), coincides with the product σ-algebra. Any probability measure on the product space with this σ-algebra is quasi-separable in the sense that, for any union of open balls that has full measure, there is a countable sub-union that also has full measure. With suitably adapted definitions, the topology of weak convergence on the space of such measures is equivalent to the topology induced by the Prohorov metric. The projection mapping from such measures to sequences of measures on the first l factors, l = 1; 2; ...; is a homeomorphism if the range of this mapping is also given a uniform metric. These findings are relevant for the theory of games of incomplete information, where a topology on the space of belief hierarchies that is based on a uniform metric has been proposed as being more appropriate for capturing the continuity properties of strategic behaviour.