On weak compactness in the space of Radon measures

Abstract

Let X be a fixed compact space, C(X) the Banach lattice of real continuous functions on X, L(X) its dual, and M(X) its bidual. [L(X) is an (L) space and M(X) an (M) space, hence the notation.] We have two objects in the present paper. The first is to give a unified development of most of the characterizations of weakly compact sets in L(X)-hence also in -F(p), p EL(X)-using purely vector lattice tools (i.e., with no measure and integration theory). There has been a long succession of characterizations, beginning with the Dunford-Pettis theorem [S] and culminating in the remarkable theorem of Grothendieck [6]. We derive them all from a theorem of Nakano [12, section 281 on the dual EC of a vector lattice E, where EC consists of the linear functionals on E which are continuous with respect to order convergence in E. While L(X) [resp 9’(p)] is not the norm-dual of M(X) [resp Sm(p)], it is precisely the set of linear functionals on M(X) [resp J?“(p)] continuous with respect to order convergence, and it is this fact which we consider to be the basis of all the weak compactness properties. Our second object stems from the above theorem of Grothendieck, and from a theorem of DieudonnC [4], which he uses. These two theorems single out a very small subspace of M(X)-in some sense, the smallest subspace containing C(X). It is a proper subspace of the first Baire class Bar, and for want of a better name we denote it by Ba1j2. The above two theorems indicate that Bali2 bears study as a space in its own right, and Part II of the paper initiates such a study.