Non-Measurable Sets

Abstract

Up to now, we have given no indication that the class of measurable sets, or of sets having the property of Baire, does not include all subsets of the line. We know that any set obtained as the result of countably many applications of union, intersection, or complementation, starting from a countable family of closed, open, or nullsets, will be measurable. It can also be shown that any analytic set is measurable. (An analytic set is one that can be represented as the continuous image of a Borel set.) According to a result of Godel [18, p. 388], the hypothesis that there exists a non-measurable set that can be represented as the continuous image of the complement of some analytic set is consistent with the axioms of set theory, provided these axioms are consistent among themselves. No actual example of a non-measurable set that admits such a representation is known (but see [40, p. 17]). Nevertheless, with the aid of the axiom of choice it is easy to show that non-measurable sets exist. We shall consider several such constructions.