Strongly meager sets and subsets of the plane

Let X ⊆ 2 . Consider the class of all Borel F ⊆ X × 2 with null vertical sections Fx, x ∈ X. We show that if for all such F and all null Z ⊆ X, ⋃ x∈Z Fx is null, then for all such F , ⋃ x∈X Fx 6= 2 . The theorem generalizes the fact that every Sierpiński set is strongly meager and was announced in [P]. A Sierpiński set is an uncountable subset of 2 which meets every null (i.e., measure zero) set in a countable set. Such sets may not exist, but they do, e.g., under the Continuum Hypothesis. A strongly meager set is a subset of 2 whose complex algebraic sum with null sets cannot give 2. Answering a question of Galvin I proved in [P] that every Sierpiński set is strongly meager (see [M] and [P] for more about Galvin’s question). Since Sierpiński sets may not exist, this result is somewhat defective. Here we prove its “absolute” version, which seems to be of independent interest. The paper is an elaboration of the Note given at the end of [P]. A different elaboration, using “small sets” of [B] and closely following my lecture at Cantor’s Set Theory meeting, Berlin 1993, is given in [BJ] in Section 5 (repeated in [BJ1]). There is, however, a major gap in the exposition in [BJ], namely, in the proof of Lemma 5.5, where one really needs a sort of Kunugui–Novikov theorem (see the proof of Lemma 4 below). Also it seems reasonable to avoid “small sets” because they do not form an ideal. Let X ⊆ 2. Consider the class of all Borel F ⊆ X×2ω with null vertical sections Fx, x ∈ X. If for all such F , ⋃ x∈X Fx is null, resp. 6= 2, we say that X ∈ Add, resp. X ∈ Cov. (See [PR] for an explanation of this notation.) Here Borel means relatively Borel. It is useful to remember that for a Borel F ⊆ X×2ω, the function x 7→ μ(Fx) is Borel. In particular, any Borel 1991 Mathematics Subject Classification: 04A15, 03E15. Partially supported by KBN grant 2 P03A 011 09.