 The Vitali theorem, which follows from the axiom, states that there exist subsets of R that are not Lebesgue-measurable.Assuming the axiom of choice, non-measurable sets with many surprising properties have been demonstrated, such as those of the Banach–Tarski paradox.In 1970, Robert M. Solovay showed that the existence of sets that are not Lebesgue-measurable is not provable within the framework of Zermelo–Fraenkel set theory in the absence of the axiom of choice (see Solovay's model).