1 5 Ja n 19 93 All meager filters may be null

We show that it is consistent with ZFC that all filters which have the Baire property are Lebesgue measurable. We also show that the existence of a Sierpinski set implies that there exists a nonmeasurable filter which has the Baire property. The goal of this paper is to show yet another example of nonduality between measure and category. Suppose that F is a nonprincipal filter on ω. Identify F with the set of characteristic functions of its elements. Under this convention F becomes a subset of 2 and a question about its topological or measure-theoretical properties makes sense. It has been proved by Sierpinski that every non-principal filter has either Lebesgue measure zero or is nonmeasurable. Similarly it is either meager or does not have the Baire property. In [T] Talagrand proved that Theorem 0.1 There exists a measurable filter which does not have the Baire property. The author thanks the Lady Davis Fellowship Trust for full support Supported by the Israel Academy of Sciences (Basic Research Fund) Publication 434