A Simple Proof for the Jordan Measurability of Convex Sets
A Simple Proof for the Jordan Measurability of Convex Sets
复制标题
凸集乔丹可测性的简单证明
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发表时间：
1997
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通讯作者：
L. Szabó
中科院分区：
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作者：
L. Szabó
V(B) = (b1−a1)（b2−a2）···（bn−an）若K是一个有界集合，令V−(K) = sup∑m V(Bm)，其中K中的所有包被有限族{B1，B2，…}的块，令V+(K) = inf∑m V(Bm)，其中极小值被有限族{B1，B2，…}块。让我们回忆一下，集合中的填充是这样一种排列，它的所有成员都包含在集合中并且具有互不相交的内部，集合的覆盖是这样一种排列，它的并集包含该集合。显然，V−(K)≤V+(K)。现在，我们说，当V−(K) = V+(K)时，有界集合K是约旦可测的，在这种情况下，我们称这个公值为K的体积。要了解更全面的情况，我们建议读者参阅专著[1]。本文的目的是给出一个合理简单的几何证明（即不使用紧性论证）
V(B) = (b1 − a1)(b2 − a2) · · · (bn − an) in the obvious way. If K ⊆ En is a bounded set, let V−(K) = sup ∑ m V(Bm), where the supremum is taken over all packings in K by finite families {B1,B2, . . .} of blocks, and let V+(K) = inf ∑ m V(Bm), where the infimum is taken over all coverings of K by finite families {B1,B2, . . .} of blocks. Let us recall that a packing in a set is an arrangement whose members are all contained in the set and have mutually disjoint interiors, and a covering of a set is an arrangement whose union contains the set. It is clear that V−(K) ≤ V+(K). Now, we say that the bounded set K ⊆ En is Jordan measurable if V−(K) = V+(K), and in this case we call this common value the volume of K . For a more comprehensive account we refer the reader to the monograph [1]. The aim of this paper is to give a reasonably simple geometric proof (i.e. without using compactness arguments) for the following well-known