# ON THE LEBESGUE SPACE OF VECTOR MEASURES

• Received : 2009.12.11
• Published : 2011.07.31
• 34 6

#### 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 $f{\in}L^1$(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

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