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Abstract. Let X be a zero-dimensional compact metrizable space endowed with a strictly positive continuous Borel σ-additive measure µ which is good in the sense that for any clopen subsets U, V ⊂ X with µ(U ) < µ(V ) there is a clopen set W ⊂ V with µ(W ) = µ(U ). We study σ-ideals with Borel base on X which are invariant under the action of the group Hµ(X) of measure-preserving homeomorphisms of (X, µ), and show that any such σ-ideal I is equal to one of seven σ-ideals:Here [X] ≤κ is the ideal consisting of subsets of cardiality ≤ κ in X, M is the ideal of meager subsets of X, N = {A ⊂ X : µ(A) = 0} is the ideal of null subsets of (X, µ), and E is the σ-ideal generated by closed null subsets of (X, µ).

Abstract. In this paper we study a notion of a κ-covering in connection with Bernstein sets and other types of nonmeasurability. Our results correspond to those obtained by Muthuvel in [7] and Nowik in [8]. We consider also other types of coverings. Definitions and notationIn 1993 Carlson in his paper [3] introduced a notion of κ-coverings and used it for investigating whether some ideals are or are not κ-translatable. Later on κ-coverings were studied by other authors, e.g. Muthuvel (cf. [7]) and Nowik (cf.[8], [9]). In this paper we present new results on κ-coverings in connection with Bernstein sets. We also introduce two natural generalizations of the notion of κ-coverings, namely κ-S-coverings and κ-I-coverings.We use standard set-theoretical notation and terminology from [1]. Recall that the cardinality of the set of all real numbers R is denoted by c. The cardinality of a set A is denoted by |A|. If κ is a cardinal number then