Probability Measures on Product Spaces with Uniform Metrics

Abstract

The paper provides mathematical foundations for a homeomorphism theorem à la Mertens and Zamir (1985) when the space of belief hierarchies of an agent has the uniform topology rather than the product topology. The Borel σ-algebra for the uniform topology being unsuitable, the theorem relies on the product σ-algebra but defines the topology of weak convergence on the space of measures on this σ-algebra with reference to the uniform topology on the underlying space. For a countable product of complete separable metric spaces, the paper shows that this topology on the space of measures on the product σ-algebra is metrizable 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.

Keywords: Product spaces with uniform metrics, weak convergence of non-Borel measures, σ-algebras generated by the open balls, quasi-separable measures, Prohorov metric

JEL Classification: C02, C72