## Illinois Journal of Mathematics

### 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.

#### Article information

Source
Illinois J. Math., Volume 57, Number 1 (2013), 145-169.

Dates
First available in Project Euclid: 23 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1403534490

Digital Object Identifier
doi:10.1215/ijm/1403534490

Mathematical Reviews number (MathSciNet)
MR3224565

Zentralblatt MATH identifier
1298.28027

#### Citation

Khan, M. Ali; Sagara, Nobusumi. Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces. Illinois J. Math. 57 (2013), no. 1, 145--169. doi:10.1215/ijm/1403534490. https://projecteuclid.org/euclid.ijm/1403534490

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