Measurability Properties of Sets of Vitali's Type

Assume a group G acts on a set. Given a subgroup H of G , by an //-selector we mean a selector of the set of all orbits of H . We investigate measurability properties of //-selectors with respect to G-invariant measures. Let us fix a set A and a group G acting on it. By p we denote a G-invariant countably additive measure on A . The most common example of such a situation is an invariant measure on a group acting on itself by translations. Let H be a subgroup of G. By an H-selector (sometimes called a set of Vitali's type for H) we understand a set having exactly one point in common with each orbit of H. Measurability properties of selectors were first systematically studied by Cichon, Kharazishvili, and Weglorz in [1]. Selectors are extremely useful in constructing sets nonmeasurable with respect to an invariant measure. The first example of a Lebesgue nonmeasurable set, due to Vitali [8], is just a Q-selector where Q is the group of rationals. Also for any finite invariant diffused measure on a group (acting on itself by translations) any //-selector for a countable subgroup H is nonmeasurable. In fact, in both cases above the constructed sets are nonmeasurable with respect to any invariant extension of a given measure. Kharazishvili in [3] and Erdos and Mauldin in [2] constructed a nonmeasurable set for any cr-finite invariant measure. Their example is the union of a family of //-selectors where H is a subgroup of cardinality cox . Strengthening the result from [2, 3] the author constructed in [6] sets nonmeasurable with respect to any invariant extension of a given CT-finite measure. These sets are subsets of //-selectors for an appropriately chosen countable group H. In the present paper we take a closer look at measurability properties of selectors. Putting a freeness assumption on the action of G and assuming that G is uncountable we prove that for a rr-finite measure one can always find a countable group H such that no //-selector is measured by any invariant extension of the given measure. We show also that the situation for subgroups of full cardinality is just the opposite. Imposing a stronger freeness condition and Received by the editors January 27, 1992 and, in revised form, March 19, 1992. 1991 Mathematics Subject Classification. Primary 28C10, 04A20; Secondary 43A05.