Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces

Abstract

This paper offers a sufficient condition, based on Maharam (Proc. Natl. Acad. Sci. USA 28 (1942) 108–111) and re-emphasized by Hoover and Keisler (Trans. Amer. Math. Soc. 286 (1984) 159–201), for the validity of Lyapunov’s theorem on the range of a nonatomic vector measure taking values in an infinite-dimensional Banach space that is not necessarily separable nor has the Radon–Nikodym property (RNP). In particular, we obtain an extension of a corresponding result due to Uhl (Proc. Amer. Math. Soc. 23 (1969) 158–163). The proposed condition is also shown to be necessary in the sense formalized by Keisler and Sun (Adv. Math. 221 (2009) 1584–1607), and thereby closes a question of long-standing as regards an infinite-dimensional generalization of the theorem. The result is applied to obtain short simple proofs of recent results on the convexity of the integral of a set-valued function, and on the characterization of restricted cores of a saturated economy.