Ramsey, Lebesgue, and Marczewski Sets and the Baire Property

We investigate the completely Ramsey, Lebesgue, and Marczewski σ-algebras and their relations to the Baire property in the Ellentuck and density topologies. Two theorems concerning the Marczewski σ-algebra (s) are presented. Theorem. In the density topology D, (s) coincides with the σ-algebra of Lebesgue measurable sets. Theorem. In the Ellentuck topology on [ω] , (s)0 is a proper subset of the hereditary ideal associated with (s). We construct an example in the Ellentuck topology of a set which is first category and measure 0 but which is not Br-measurable. In addition, several theorems concerning perfect sets in the Ellentuck topology are presented. In particular, it is shown that there exist countable perfect sets in the Ellentuck topology. 0. Introduction. We are interested in the σ-algebras B of Borel sets, L of Lebesgue measurable sets, (s) of Marczewski measurable sets, Bw of sets with the Baire property in the wide sense, Br of sets with the Baire property in the restricted sense, and CR of sets which are completely Ramsey. B, Bw, Br, and (s) have a well-defined meaning in any topological space, and we are particularly interested in the Euclidean, Ellentuck, and density topologies. 1991 Mathematics Subject Classification: Primary 28A20, 26A15; Secondary 04A15, 54C30.