On Additive Almost Continuous Functions under Cpa

We prove that the Covering Property Axiom CPA prism, which holds in the iterated perfect set model, implies that there exists an additive discontinuous almost continuous function f : R → R whose graph is of measure zero. We also show that, under CPA prism, there exists a Hamel basis H for which, E+(H), the set of all linear combinations of elements from H with positive rational coefficients, is of measure zero. The existence of both of these examples follows from Martin’s axiom, while it is unknown whether either of them can be constructed in ZFC. As a tool for the constructions we will show that CPA prism implies its seemingly stronger version, in which ω1-many games are played simultaneously. 1. Preliminaries and axiom CPA prism Our set theoretic terminology is standard and follows that of [1]. In particular, |X| stands for the cardinality of a set X and c = |R|. The Cantor set 2 will be denoted by a symbol C. We use term Polish space for a complete separable metric space without isolated points. For a Polish space X symbol Perf(X) will stand for a collection of all subsets of X homeomorphic to the Cantor set C. For a fixed 0 < α < ω1 and 0 < β ≤ α a symbol πβ will stand for the projection from C onto C . In what follows we will consider R as a linear space over Q. For Z ⊂ R its linear span with respect to this structure will be denoted by LIN(Z). A subset H of R is a Hamel basis provided it is a linear basis of R over Q, that is, it is linearly independent and LIN(H) = R. Axiom CPA prism was introduced by the authors in [5], where it is shown that it holds in the iterated perfect set model. Also, CPA prism is a simpler version of the axiom CPA which is described in a monograph [9]. For the reader’s convenience, we will restate CPA prism in the next few paragraphs. For 0 < α < ω1 let Φprism(α) be the family of all continuous injections f : C → C with the property that f(x) β = f(y) β ⇔ x β = y β for all β ∈ α and x, y ∈ C. Functions Φprism(α) are called projection-keeping homeomorphisms. (Compare [11].) Let Pα = {range(f) : f ∈ Φprism(α)} and Pω1 = ⋃ 0<α<ω1 Pα. We will refer to elements of Pω1 as iterated perfect sets. (In [17] the elements of Pα are called I-perfect, 1991 Mathematics Subject Classification. Primary 26A15, 26A30; Secondary 03E35.