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.