Null sets with respect to a continuous function

Abstract

This short paper concerns \lq\lq peso nullo'' subsets of the real line defined by Renato Caccioppoli \cite{ca}. The framework is that of integration with respect to a function $g$ which is continuous but not necessarily of bounded variation. Here we shall call these sets $g$-null. Since the family of $g$-null sets is a $\sigma$-ideal, the natural question is whether it is a family of null sets with respect to a Borel measure on the real line. The paper gives a negative answer to this question.