Consider an arbitary point set S on the real number line or in Euclidean n-space (the limitation to Euclidean space is unessential). The set S has an interior Lebesgue measure mi(S) and an exterior Lebesgue measure me(S). There are the following well known inequalities for two disjoint sets S1 and S2: mi(S1US2) [unk] mi(S1) + mi(S2), me(S1US2) ≤ me(S1) + me(S2). These state the superadditivity of interior measure for disjoint sets, and the subadditivity of exterior measure. The question is posed as to what conditions, besides these, are there on the six quantities mi() and me() for the disjoint sets S1,S2 and for their union S1US2.The present paper obtains a complete set of conditions on these six quantities. These are in the form of inequalities. The variability can be expressed in terms of six independent nonnegative quantities, and the six quantities mi() and me() for the disjoint sets S1, S2 and for S1US2 can be expressed as suitable sums of these.There appears the interesting property that the „average measure” ma(S), defined by ma(S) = (mi(S) + me(S))/2, is subadditive. Also, any linear combination αmi(S) + βme(S) is discussed.
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机译：考虑实数线上或欧几里德n空间中的任意点集S（对欧几里德空间的限制是不必要的）。集合S具有内部Lebesgue度量mi（S）和外部Lebesgue度量me（S）。两个不相交集S1和S2存在以下众所周知的不等式：mi（S1US2）[unk] mi（S1）+ mi（S2）， me （ S 1U S 2）≤ me （ S 1）+ m e （ S 2 ）。这些陈述了不相交集的内部度量的超可加性和外部度量的 subadditivity 。提出的问题是，除了这些条件，六个量 m i （）和 m e < / em>（）用于不相交集 S 1 ， S 2 及其联合 S 1 U S 2 。本文针对这六个量获得了一个完全条件集。这些形式为不等式。可以用六个独立的非负量以及六个量 m i （em）和 m e 来表示变异性。 em>（）用于不相交集 S 1 ， S 2 和 S 1 U S 2 可以表示为这些值的合适和。似乎有一个有趣的性质，即“平均度量” m a （ S ），由 m a （ S ）=（ m i （ S ）+ m e （ S ））/ 2，是次加性。此外，任何线性组合α m i （ S ）+β m e （ S ）。
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