Null sets with respect to a continuous function

This short paper concerns “peso nullo” subsets of the real line defined in [Caccioppoli, R.: L’integrazione e la ricerca delle primitive rispettoad una funzione continua qualunque, Ann. Mat. Pura Appl. 40 (1955), 15-34]. 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 σ-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.