A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable

Abstract

We show that the existence of a non-Lebesgue measurable set cannot be proved in Zermelo-Frankel set theory (ZF) if use of the axiom of choice is disallowed. In fact, even adjoining an axiom DC to ZF, which allows countably many consecutive choices, does not create a theory strong enough to construct a non-measurable set. Let ZFC be Zermelo-Frankel set theory together with the axiom of choice. Let I be the statement: There is an inaccessible cardinal'.