Measure-Theoretic Properties of Arbitrary Point Sets
AUTOR(ES)
Shiffman, Max
RESUMO
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.
