Finitely Extendable Functionals on Vector Lattices

A general extended functional is a functional which is allowed to take on infinite values; in other words such functionals are similar to the Lebesgue integral on the space of all integrable functions. The problem of representing a general extended functional f on a vector lattice X as an operator with finite values only has been solved in 12]. In fact, this problem has been solved for a larger class of general extended operators on an ordered vector space. The solution of this problem was given by means of an extension of the range R of the functional f to some ultrapower of R. Notice, however, that it is not always the case that a functional f can be considered as a trace of some internal functional *f:*XR. (We remark, without going into details, that such an internal functional exists exactly in the case when the results of the nonstandard analysis can be used for investigation of the given functional.) In 12] a standard necessary and sufficient condition was given for solving this latter problem on the existence of *f. Namely, f is a trace of an internal *f if and only if it is finitely extendable. This result makes the finite extendability problem worthy of study. The present paper is a first attempt in this direction. Simultaneously we introduce and study a more general notion of weak finite extendability that coincides with finite extendability, for instance, for vector lattices with a strong unit.