ABSTRACT
This chapter establishes some interesting connections between absolutely nonmeasurable functions and the measure extension problem. It shows that the existence of absolutely nonmeasurable solutions of Cauchy’s functional equation can be proved under Martin’s Axiom. In this context, any nontrivial solution of Cauchy’s functional equation is necessarily nonmeasurable with respect to the classical Lebesgue measure. The chapter obtains a certain generalization of the result of Pelc and Prikry concerning the measure extension problem. It also includes exercise problems related to absolutely nonmeasurable functions.
