Measure-theoretic properties of non-measurable sets (Q1096031)

Considering point sets of the real number line or in Euclidean space (the limitation to Euclidean space is unessential; general measure spaces, with a limitation, can also be considered), this article discusses the interior and exterior measures of two disjoint point sets \(S_ 1\), \(S_ 2\) and their union set \(S_ 1\cup S_ 2.\) Besides well-known inequalities on the six quantities \(m_ i(S)\) and \(m_ e(S)\) for \(S=S_ 1,S_ 2,\) and \(S_ 2\cup S_ 2,\) further inequalities are obtained. Indeed, a complete collection of inequalities on these six quantities is obtained, which are both necessary and sufficient conditions. One of the inequalities can be expressed as: introducing the average measure, defined by \(m_ a(S)=(m_ i(S)+m_ e(S)),\) then \(m_ a(S_ 1\cup S_ 2)\leq m_ a(S_ 1)+m_ a(S_ 2),\) i.e., average measure is subadditive. The complete collection of inequalities are expressible as: there are a certain six linear combinations of the six quantities which are each \(\geq 0,\) and these six linear combinations can be independently assigned any non-negative real value or \(\infty\), subject to their sum being \(\leq m(X),\) where X is the entire space or a measurable set containing \(S_ 1\) and \(S_ 2\).