Some applications of partitioning covers (Q1068225)

Let \(S_ 0\) be the class of all closed subintervals of \(I_ 0=[a,b].\) Suppose that for each \(x\in I_ 0\) there is a system \(S_ x\subset S_ 0\) with the following property: For each \(I\subset I_ 0\) there are \(x_ 1,x_ 2,...,x_ n\in I_ 0\) and \(I_ j\in S_{x_ j}\) such that \(I_ 1,I_ 2,...,I_ n\) do not overlap and \(I=\cup I_ j.\) Then the family \(\bar S=\cup S_ x,\) \(x\in I_ 0,\) is called a partitioning cover of \(I_ 0\). A family \(S\subset S_ 0\) is said to be additive if \([c,d],[d,e]\in S\) implies \([c,e]\in S.\) S is said to be \(\bar S-\)local if \(\bar S\subset S.\) Lemma: If a family S is both additive and \(\bar S- \)local for some partitioning cover \(\bar S,\) then \(S=S_ 0.\) This simple lemma makes some proofs of theorems on covering, monotonicity and absolute continuity very simple and direct.