Abstract. With a measure $\varphi $ on a $\sigma $-algebra $\Sigma $ of sets taking values in a Banach space two positive functions on $\Sigma $, called semivariations of $\varphi $, are associated. We characterize those functions as order continuous submeasures that are multiply subadditive in the sense of G. G. Lorentz (1952). In connection with some results of G. Curbera (1994) and the author (2003), we also discuss the special cases where $\varphi $ is separable and nonatomic or has relatively compact range.

AMS Subject Classification (1991): 28B05, 46G10, 28A12

Keyword(s): Banach space, vector measure, semivariation, relatively compact range, separable, nonatomic, submeasure, order continuous, multiply subadditive

Received March 23, 2009, and in revised form December 10, 2009. (Registered under 45/2009.)