This article discusses the
interior and exterior measures of two disjoint point sets S1, S2 and their union
set S1∪ S2. Besides well-known inequalities on the six quantities mi(S)
and me(S) for S = S1,S2, and S1∪ S2, further inequalities are obtained.
Indeed, a complete colleciton of inequalities on these six quantities is obtained,
which are both necessary and sufficient conditions. The complete collection of
inequalities are expressible as: there are a certain six linear combinations of the six
quantities which are each ≥ 0, and these six linear combinations can be
independently assigned any nonnegative real value or ∞, subject to their sum being
≤ m(X), where X is the entire space or a measurable set containing S1 and
S2.
