  Consider a measurable space with an atomless finite vector measure. This
measure defines a mapping of the $\sigma$-field into an Euclidean space.
According to the Lyapunov convexity theorem, the range of this mapping is a
convex compactum. Similar ranges are also defined for measurable subsets of the
space. Two subsets with the same vector measure may have different ranges. We
investigate the question whether, among all the subsets having the same given
vector measure, there always exists a set with the maximal range of the vector
measure. The answer to this question is positive for two-dimensional vector
measures and negative for higher dimensions. We use the existence of maximal
ranges to strengthen the Dvoretzky-Wald-Wolfowitz purification theorem for the
case of two measures.
