On the Lebesgue space of vector measures
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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∈ 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∈ 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.
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∈ 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∈ 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.
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