Hamel-isomorphic images of the unit ball Jacek Cicho´ n and PrzemyslawSzczepaniak Abstract. In this article we consider linear isomorphisms over the field of rational numbers between linear spaces R 2 and R. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly non-measurable subset of the real line, which is strongly non-similar to any classical non- measurable subset of reals. We also show the consistency and independence of the proposition that all images of bounded measurable subsets of the plane via a such mapping are non-measurable. 1. Introduction Let us recall a well known theorem essentially due to S. Banach. Let (G, +) and (H, +) be locally compact Polish groups (not necessarily abelian). If f : G −→ H is a homomorphism, which is Haar measurable or has the Baire property, then f is continuous. The proof follows immediately from well known theorem of H. Steinhaus. Indeed, if f has one of the above properties, then there exists a ”massive” set A ⊆ G such that f ↾ A is continuous. Then f ↾ A − A is also continuous and by that theorem of Steinhaus A − A contains a neighborhood of unity. Thus f is continuous everywhere. Let R denotes the real line. Let us say that X ⊆ R m is strictly non-measurable if the inner measures of X and of R m \ X both vanish. And X is strictly non-Baire if all Borel sets included in X or in R m \ X are meager (i.e. of the first category). Consider the case when G = R m , H = R and f is discontinuous. Then if I is a non-degenerated interval, then f -1 [I ] is strictly non-measurable and strictly non-Baire. This was shown by A. Ostrowski and M. Kuczma (see [13], [8]). All that suggested to us a study of images of sufficiently regular subsets of R 2 in the case when f is an isomorphism of R 2 onto R. Recall that all the spaces R m (m> 0) viewed as linear spaces over the field Q of rational numbers are isomorphic (all have Hamel bases of the same power c). Let f be an isomorphism of R 2 onto R. We shall prove that if D is a disk of positive radius, then f [D] is non-measurable and lacks the Baire property. Moreover, 1. The image f [D] is strictly non-measurable and strictly non-Baire; 2. The following proposition is consistent in ZFC: (⋆) For all isomorphisms f : R 2 −→ R and all bounded measurable sets A ⊆ R 2 of positive measure (or containing a non-meager Borel set), f [A] is not measurable (or non-Baire). I. Reclaw has proved that if 1991 Mathematics Subject Classification. Primary 03E35, 03E75; Secondary 28A99. Key words and phrases. Lebesgue measure, Baire property, null sets, non-measurable sets. 1