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Journal of Function Spaces and Applications
Volume 2 (2004), Issue 1, Pages 1-16
http://dx.doi.org/10.1155/2004/413765

Spaces of complex functions and vector measures in incomplete spaces

1Math.-Geogr. Fakultät, Katholische Universitat Eichstätt-Ingolstadt, D-85071 Eichstätt, Germany
2University of Würzburg, Department of Mathematics, Am Hubland, D-97074 Würzburg, Germany

Received 1 December 2002

Academic Editor: Jürgen Appell

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It is known that the space L1(μ) of complex functions which are integrable with respect to a vector measure μ taking values in a (not neessarily complete) locally convex space is not an ideal, in general. We discuss several natural properties which L1(μ) may or may not possess and consider various implications between these properties. For a particular class of properties, whether or not there exists a partiular space of the form L1(μ) having these properties, is shown to be equivalent to the existence of any space of complex functions on C having these same properties.