Maharam-types and Lyapunov's theorem for vector measures on Banach spaces - M. Ali Khan; Nobusumi Sagara (The Johns Hopkins University, Department of Economics, 2012)
This paper offers a sufficient condition, based on Maharam (1942) and re-emphasized by Hoover-Keisler (1984), for the validity of Lyapunov's (1940) theorem on the range of an atomless vector measure taking values in an infinite-dimensional Banach space that is not necessarily separable nor has the RNP property. In particular, we obtain an extension of a corresponding result due to Uhl (1969). The proposed condition is also shown to be necessary in the sense formalized in Keisler-Sun (2009), 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.
Keywords: 330; Saturated measure space; Lyapunov's theorem; Maharam-type; integral of correspondences; measurable selection; Radon-Nikodym property; saturated economy; restricted cores