Consistent Extensions of Linear Functionals and of Probability Measures

Suppose we are given a family of fields 3;a, a E a?, of subsets of a fixed set X, and for each a a finite measure ma defined on JF. Under what conditions will there exist a measure m, defined on a field F containing each .0, and agreeing with each ma on 9;;? This problem is of some importance in probability theory (it arises in the theory of marginals), and has been studied, for instance, in [3], [5], [7], [14], [16]. The problem takes several forms: the measures considered can be positive or signed, dominated by a previously given measure or not, and countably or finitely additive. The countably additive cases are the most significant, but also the most difficult (see [1], [4], [8]); the most far reaching results here seem to be those of Kellerer [5], [6]. However, the finitely additive cases are also of interest (in fact, the case in which X itself is finite is of significance; see [16]), and the present paper deals almost entirely with them. Our main object is to give a unified, simple, general treatment of the finitely additive cases, allowing arbitrarily many measures ma (in the literature, only finitely many are usually considered). From this we shall obtain one theorem providing a countably additive extension, under additional topological assumptions. We shall also deal with a second general problem (see [16]): under what conditions on the fields 3a will every consistent set of measures have a (finitely additive) consistent extension? It is convenient to reformulate the finitely additive problems in a slightly more general form, which reduces them to linear algebra. Let L denote the set of all real valued functions on X; for each a E X, let La be the subset of L consisting of all F. measurable step functions on X, and for each f e L., let