Extension of Lyapunov's Convexity Theorem to Subranges
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Consider a measurable space with a finite vector measure. This measure defines a mapping of the &sigma-field into a Euclidean space. According to Lyapunov's convexity theorem, the range of this mapping is compact and, if the measure is atomless, this range is convex. Similar ranges are also defined for measurable subsets of the space. We show that the union of the ranges of all subsets having the same given vector measure is also compact and, if the measure is atomless, it is convex. We further provide a geometrically constructed convex compact set in the Euclidean space that contains this union. We show that, for two-dimensional measures, among all the subsets having the same given vector measure, there exists a set with the maximal range of the vector measure (maximal subset). Furthermore, for two-dimensional measures, the maximal subset, the above-mentioned union, and the above-mentioned convex compact set are equal sets. We also give counterexamples showing that, in three or higher dimensions, the maximal subset may not exist and these equalities may not hold. We use the existence of maximal subsets to strengthen the Dvoretzky-Wald-Wolfowitz purification theorem for the case of two measures. We show that there are no similar results for three or higher dimensions.