Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 57, Number 1 (2013), 145-169.
Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces
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.
Illinois J. Math., Volume 57, Number 1 (2013), 145-169.
First available in Project Euclid: 23 June 2014
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 28B05: Vector-valued set functions, measures and integrals [See also 46G10] 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22] 28B20: Set-valued set functions and measures; integration of set-valued functions; measurable selections [See also 26E25, 54C60, 54C65, 91B14] 46B22: Radon-Nikodým, Kreĭn-Milman and related properties [See also 46G10]
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