A mixed measure of content on the set of real numbers

摘要

The counting measure applies only on countable subsets of the set of real numbers. On the other hand, the Lebesgue measure applies on any countable union of intervals but ignores countable subsets since it assigns to them a null weight indiscriminately. This paper proposes a measure of content which applies on finite unions of intervals and enables to differentiate finite sets. This measure of content is shown to be a Choquet capacity. Furthermore, extension onto the system of all subsets of the real number set is discussed and ideas for generalization to the multidimensional space are presented. A class of content-based measures of comparison is also suggested, along with a discussion of some of their basic properties.