Abstract

A positive and normalised real linear functional on the set of bounded continuous functions can be characterised as the integral of a σ-additive probability measure, by the F. Riesz Representation Theorem. In this paper, we look at the finitely additive extensions of such a functional to the set of all bounded random variables, and prove that they are determined by Riesz’ extension to lower semi-continuous functions. In doing so, we establish links with Daniell’s approach to integration, Walley’s theory of coherent lower previsions, and de Finetti’s Representation Theorem for exchangeable random variables.KeywordsF. Riesz representation theoremLower semi-continuityCoherent lower previsionNatural extensionI-integralExchangeability

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