International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 1, Pages 21-30

Bounded sets in the range of an X-valued measure with bounded variation

B. Marchena and C. Piñeiro

Departamento de Matemáticas, Escuela Politécnica Superior, Universidad de Huelva, La Rábida, Huelva 21810, Spain

Received 20 July 1998

Copyright © 2000 B. Marchena and C. Piñeiro. 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.


Let X be a Banach space and AX an absolutely convex, closed, and bounded set. We give some sufficient and necessary conditions in order that A lies in the range of a measure valued in the bidual space X and having bounded variation. Among other results, we prove that X is a G. T.-space if and only if A lies inside the range of some X-valued measure with bounded variation whenever XA is isomorphic to a Hilbert space.