Concentrated and rarified sets of points

Abstract

I have arrived at the definition of concentra ted sets f rom the following two similar problems: P rob lem I. What are the linear sets whose measure with respect to any continuous monotone function q~ (1) is zero? Problem II . What are the linear sets on which the variation of any continuous monotone function is zero? Concentra ted sets are defined as follows: A non-enumerable set of points E is said to be concentrated in the neighbourhood of an enumerable set H i f any open set containing the set H contains also the set E with the exception of at most an enumerable set of points.