Integral representations of weakly compact operators

Abstract

Let X be a completely regular space and JB, F locally convex spaces. Denote by C rc = C rc (X, E) the space of all continuous functions / from X into E for which f(X) is relatively compact. Uniformly continuous weakly compact operators from Crc into F are represented by integrals with respect to SP(E, F) valued measures on the algebra generated by the zero sets. Necessary and sufficient conditions for an operator to be continuous, with respect to certain topologies, are obtained. A sufficient condition for extending a measure to all Baire sets is given.