We prove that the Covering Property Axiom CPA_prism^game, which holds in the iterated perfect set model, implies that there exists an additive discontinuous almost continuous function f from R to R whose graph is of measure zero. We also show that, under CPA_prism^game, there exists a Hamel basis H for which the set E⁺(H), 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^game implies its seemingly stronger version, in which \omega₁-many games are played simultaneously.

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Last modified February 2, 2006.