李雅普诺夫凸性定理对子丛的推广
Extension of Lyapunov's Convexity Theorem to Subranges
摘要：
考虑具有有限向量测度的可测空间。该测度定义了$σ$域到欧几里得空间的映射。根据李雅普诺夫凸性定理，这个映射的值域是紧的，如果测度是无原子的，这个值域是凸的。对于空间的可测量子集也定义了类似的范围。我们表明，具有相同的给定向量测度的所有子集的范围的工会也是紧凑的，如果措施是atomless，它是凸的。我们进一步在包含这个并集的欧几里得空间中给出了一个几何构造的凸紧集。这两个集合的相等性在二维测度中成立，但在更高的维度中可能会被违反。
英文摘要：
Consider a measurable space with a finite vector measure. This measure defines a mapping of the $σ$-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 compactum in the Euclidean space that contains this union. The equality of these two sets, that holds for two-dimensional measures, can be violated in higher dimensions.
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