Some applications of partitioning covers
type: Item
instance of: scholarly article
title: Some applications of partitioning covers
zbMATH Open document ID: 0581.26010
published in: Real Analysis Exchange
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 26A48
Mathematics Subject Classification ID: 26A46
zbMATH DE Number: 3929345
zbMATH Keywords: covering
zbMATH Keywords: monotonicity
zbMATH Keywords: absolute continuity
author: P. S. Bullen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3929345