ON THE LEBESGUE SPACE OF VECTOR MEASURES

Abstract

In this paper we study the Banach space L(1)(G) of real valued measurable functions which are integrable with respect to a vector measure G in the sense of D. R. Lewis. First, we investigate conditions for a scalarly integrable function f which guarantee f is an element of L(1)(G). Next, we give a sufficient condition for a sequence to converge in L(1)(G). Moreover, for two vector measures F and G with values in the same Banach space, when F can be written as the integral of a function f is an element of L(1)(G), we show that certain properties of G are inherited to F; for instance, relative compactness or convexity of the range of vector measure. Finally, we give some examples of L(1)(G) related to the approximation property.