A lemma on maximal sets and the theorem of Denjoy-Vitali

Abstract

By an ∮-related family ∮ we mean a non-empty family ∮ of elements such that to each element F ∈ ∮ is associated a set R(F) of elements of ∮, called the R-class of F, which contains F. An element G ∈ R(F) is said to be R-related to F. By an R-section S of ∮ we mean a set of elements of ∮ such that for any elements F1, F2 of S either F1 ∈ R(F2) or F2 ∈R(F1). If R(F) = {F} for each F ∈ ∮ then the only R-Sections are the sets {F}. The interesting applications of the lemma proved below are to those cases when there exist R-sections which do not contain a finite number of elements.