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Probability measures on product spaces with uniform metrics


Hellwig,  Martin
Max Planck Institute for Research on Collective Goods, Max Planck Society;

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Hellwig, M. (2017). Probability measures on product spaces with uniform metrics.

Cite as: https://hdl.handle.net/11858/00-001M-0000-002D-2FA8-2
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.