We develop some ideas contained in the author's paper  which was, in turn, inspired by Bierlein and Stich . The main body of the present paper is divided into three sections. Section 2 is concerned with some vector-lattice-theoretical results. They are then applied to extensions of quasi-measures and measures in Sections 3 and 4, respectively.
Let X be a vector lattice, let x ε X+ and let S be a non-empty set. Theorems 1 and 2 describe some properties of the convex set
(see Section 2 for the definition of the sum above). The extreme points of Dx,s are characterized in terms of the components of x. It is also shown that if X has the principal projection property and S is countable, then extr Dx,s is, in some sense, large in Dx,s. Furthermore, for finite S, each point in Dx,s is then a sσ-convex combination of extreme ones.