Generalized Measures Whose Values Are Operators into an Intermediate Space by Gregers L. Krabbe

The object of this note is the announcement of some results involving integration on [ — <», <*> ] with respect to functions whose values are operators into an intermediate space (see Definition B). Besides several other function spaces of interest in analysis, Sobolev and Lebesgue spaces are particular types of intermediate spaces (cf., the recent contributions of CALDERÓN, GAGLIARDO, KREIN, LIONS) ; consequently, the present results extend previous ones dealing with Lebesgue spaces L(a, Cfc, ju) (cf. KRABBE). Proofs and further theorems will appear elsewhere. Each integrand will belong to the space Go of regulated functions on the interval (— oo, oo). Let [Go] consist of all g £ G o such that g coincides with its right-hand limit at each point; a Banach space [Go] is obtained by endowing [G0] with the topology of uniform convergence. Let Gi be the family of functions of bounded variation on ( — oo , oo ) ; in Definition C we shall characterize a strictly-decreasing sequence {GT: 0 < r < l } of topological spaces such that