Is it possible to construct non measurable sets....?

Question

In R^n (with respect to the Lebesgue measure) without using the Axiom of Choice? All the examples I've seen, like Vitali's set, are constructed resorting to the Axiom of Choice. If AC is mandatory, then does this mean non measurable sets have a, say, virtual but not actual existence?
Thank you.

Answer 1

No. The Axiom of Choice must be presumed in order to construct or prove the existence of nonmeasurable sets. This was proven by Robert M. Solovay in 1970, I believe in this paper:

Solovay, Robert M. (1970). "A model of set-theory in which every set of reals is Lebesgue measurable". Annals of Mathematics. Second Series 92: 1-56.