  The purpose of this paper is twofold. First, we axiomatize preference
relations on a $\sigma$-algebra of a saturated measure space represented by a
vector measure and furnish a utility representation in terms of a nonadditive
measure satisfying the appropriate requirement of continuity and convexity.
Second, we investigate the fair division problems in which each individual has
nonadditive preferences on a $\sigma$-algebra invoking our utility
representation result. We show the existence of individually rational Pareto
optimal partitions, Walrasian equilibria, core partitions, and Pareto optimal
envy-free partitions.
