On Vitali's Theorem for Groups of Motions of Euclidean Space

Abstract

We give a characterization of all those groups of isometric transformations of a Þnite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem (1) holds true. This charac- terization is formulated in purely geometrical terms. A well-known result due to Vitali (1) states that there are subsets of the real line R, nonmeasurable in the Lebesgue sense. Moreover, the argument of Vitali yields simultaneously that if X is an arbitrary Lebesgue measurable subset of R with strictly positive measure, then there exists a subset of X nonmeasurable in the Lebesgue sense. This classical result was generalized in many directions (see, e.g., (2)-(12)). In the present paper, we consider some questions relevant to the above-mentioned result of Vitali, for various groups of motions of Þnite-dimensional Euclidean spaces. The main goal of the paper is to describe those groups of motions for which an analogue of Vitali's result remains true (we recall that, in Vitali's theorem, a basic group of transformations is the group of all translations of the real line). Let E denote a Þnite-dimensional Euclidean space and let G be a sub- group of the group of all isometric transformations (i.e., motions) of E. Then the pair (E;G) can be regarded as a space equipped with a trans- formation group. We denote by K the open unit cube in E. Let n be a measure given on E and let dom(n) denote the domain of n. We shall say that n is a G-measure on E if the following two conditions are fulÞlled: (1) K 2 dom(n); n(K) = 1; (2) n is invariant with respect to G, i.e., dom(n) is a G-invariant o-ring of subsets of E and, for each set X 2 dom(n) and for each transformation g 2 G, we have n(g(X)) = n(X).