The Hahn-banach Theorem Implies the Existence of a Non Lebesgue-measureable Set

Few methods are known to construct nonLebesgue-measurable sets of reals: most standard ones start from a well-ordering of R, or from the existence of a non-trivial ultrafilter over ω, and thus need the axiom of choice AC or at least the Boolean Prime Ideal theorem (BPI see [5]). In this paper we present a new way for proving the existence of non-measurable sets using a convenient operation of a discrete group on the Euclidian sphere. The only choice assumption used in this construction is the Hahn-Banach theorem, a weaker hypothesis than BPI (see [9]). Our construction proves that Hahn-Banach theorem implies the existence of a non-measurable set of reals. This answers questions in [9], [10]. (Since we do not even use the countable axiom of choice, we cannot assume the countable additivity of Lebesgue measure; e.g. the real numbers could be a countable union of countable sets.)