On additive absolutely nonmeasurable Sierpiński-Zygmund functions.

Abstract

Assuming Martin's Axiom, it is proved that there exists a Sierpiński-Zygmund function, which is additive (i.e., is a solution of the Cauchy functional equation) and is absolutely nonmeasurable with respect to the class of all nonzero $\sigma$-finite diffused measures on the real line ${\mathbb R}$.