It is proved (Theorem 1) that for a Banach space X the following assertions are equivalent: (1) the range of every X- valued
- additive nonatomic measure of finite variation possesses a convex closure; (2) L
1 does not signembed in X.
Published in Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1192–1200, September, 1992.