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

§0. Introduction. Few methods are known to construct non Lebesgue-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 the 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.) In fact we prove (under Hahn-Banach theorem) that there is no finitely additive, rotation invariant extension of Lebesgue measure to P(R3). Notice that Hahn-Banach implies the existence of a finitely additive, isometry invariant extension of Lebesgue measure to P(R2) (see [14]). We use standard set-theoretical notation and terminology. For example, if X is any set, P(X) is the power set of X. If A ⊆ X and f : X → Y is a map, then f [A] is the image of A under f . Furthermore, ω is the set of all natural numbers. We assume ZF throughout this paper; no choice assumption (even countable) is made.