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

On Maximal Ranges of Vector Measures for Subsets and Purification of Transition Probabilities