Measure spaces

Abstract

The concept of measure of a set originates from the classical notion of volume of an interval in ℝN. Starting from such an intuitive idea, by a covering process one can assign to any set a nonnegative number which “quantifies its extent”. Such an association leads to the introduction of a set function called exterior measure, which is defined for all subsets of ℝN. The exterior measure is monotone but fails to be additive. Following Carathéodory’s construction, it is possible to select a family of sets for which the exterior measure enjoys further properties such as countable additivity. By restricting the exterior measure to such a family one obtains a complete measure. This is the procedure that allows to define the Lebesgue measure in ℝN. The family of all Lebesgue measurable sets is very large: sets that fail to be measurable can only be constructed by using the Axiom of Choice.