Bulletin of the
Korean Mathematical Society BKMS

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 $f\in L^{1}(G)$. 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 $f\in L^{1}(G)$, 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.

Keywords: Lebesgue space of vector measure, convergence in $L^{1}(G)$, the range of vector measures, Lyapunov convexity theorem, the approximation property

MSC numbers: Primary 28B10