The Carathéodory extension theorem for vector valued measures

Abstract

Abstract. This paper comprises three advertisements for a known theorem which, the author believes, deserves the title of the Caratheodory extension theorem for vector valued premeasures. Principal among these is a short and transparent proof of Porcelli's criterion for the weak convergence of a sequence in the Banach space of bounded finitely additive complex measures defined on an arbitrary field, and equipped with the total variation norm. Also, a characterization of the so-called Caratheodory Extension Property is presented, and there is a brief discussion of the relevance of this material to stochastic integration.Amongst the numerous criteria for the extendability of a vector valued premeasure (see [9] for an admirable summary and comprehensive bibliography), precisely one stands out both for its simplicity and for its applicability. Our aim in this paper is to rescue this criterion from its relative obscurity within the speciahst literature by providing three "advertisements" for it. The main advertisement will be a short, transparent proof of Porcelli's characterization of the weak convergence of a sequence of finitely additive complex measures (Theorem 2). The second advertisement will be a simple characterization of those Banach spaces for which the classical Caratheodory extension theorem remains essentially valid (Theorem 5), and the third advertisement will be an indication of the usefulness of the criterion to the construction of stochastic integrals.We begin with a brief discussion of the criterion itself. Recall that a (Banach space valued) set function u is said to be strongly bounded if we have lim"^w p(E") = 0 for every infinite sequence {£"} of pairwise disjoint sets in the domain of ¡i, and that ju is a premeasure if we have li(E)=^p(E") 71 for every finite or infinite sequence {E"} of pairwise disjoint sets in the domain of u whose union, E, also lies in the domain of u. Here now is the theorem which, we feel, deserves the title of the Caratheodory extension theorem for Banach space valued premeasures.