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On additive almost continuous functions under CPAgame/prism

100%

Ciesielski K. , Pawlikowski J.

Journal of Applied Analysis

2005

tom Vol. 11, nr 2

153-170

We prove that the Covering Property Axiom CPAgame/prism, which holds in the iterated perfect set model, implies that there ex-ists an additive discontinuous almost continuous function f : R -> R whose graph is of measure zero. We also show that, under CPAgame/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 CPAgame/prism implies its seemingly stronger version, in which ω-many games are played simultaneously.

CPAgame/prism and ultrafilters on Q and ω

86%

Millán A.

Journal of Applied Analysis

2006

tom Vol. 12, nr 2

153-180

In this paper we use the version CPAgame/prism of the Covering Property Axiom, which has been formulated by Ciesielski and Paw- likowski and holds in the iterated perfect set model, to study the rela- tions between different kinds of ultrafilters on ω and Q. In particular, we will give a full account for the logical relations between the properties of being a selective ultrafilter, a P-point, a Q-point, and an ω 1-OK point.

On some properties of Hamel bases and their applications to Marczewski measurable functions

58%

Dorais F. , Filipów R. , Natkaniec T.

Open Mathematics

2013

tom 11

nr 3

487-508

We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.