Measure-theoretic properties of arbitrary point sets.
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 m(i)(S) and an exterior Lebesgue measure m(e)(S). There are the following well known inequalities for two disjoint sets S(1) and S(2): m(i)(S(1)US(2)) [unk] m(i)(S(1)) + m(i)(S(2)), m(e)(S(1)US(2)) </= m(e)(S(1)) + m(e)(S(2)). 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 m(i)() and m(e)() for the disjoint sets S(1),S(2) and for their union S(1)US(2).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 m(i)() and m(e)() for the disjoint sets S(1), S(2) and for S(1)US(2) can be expressed as suitable sums of these.There appears the interesting property that the "average measure" m(a)(S), defined by m(a)(S) = (m(i)(S) + m(e)(S))/2, is subadditive. Also, any linear combination alpham(i)(S) + betam(e)(S) is discussed.
Shiffman
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CALIFORNIA STATE COLLEGE, HAYWARD, CALIF. 94542.