<abstract xmlns="http://www.w3.org/1999/xhtml"><p> This short paper concerns “peso nullo” subsets of the real line defined in [Caccioppoli, R.: <italic>L’integrazione e la ricerca delle primitive rispetto</italic><italic>ad una funzione continua qualunque</italic>, Ann. Mat. Pura Appl. <bold>40 </bold>(1955), 15-34]. The framework is that of integration with respect to a function <italic>g </italic>which is continuous but not necessarily of bounded variation. Here we shall call these sets <italic>g</italic>-null. Since the family of <italic>g</italic>-null sets is a <italic>σ</italic>-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.</p></abstract>

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.

Null sets with respect to a continuous function

Department of Mathematics and Statistics (Vincenzo Aversa) Department of Mathematics and Applications (Anna De Simone) University of Naples Federico II Via Cinzia I–80126-Napoli ITALY

Tatra Mountains Mathematical Publications

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This short paper concerns “peso nullo” subsets of the real line defined in [Caccioppoli, R.: L’integrazione e la ricerca delle primitive rispettoad una...

Null sets with respect to a continuous function

Published Online: Nov 19, 2012

Page range: 47 - 51

DOI: https://doi.org/10.2478/v10127-012-0024-x

Keywords
null sets, Borel measure.

This content is open access.

Null sets with respect to a continuous function

Published Online: Nov 19, 2012

Page range: 47 - 51

DOI: https://doi.org/10.2478/v10127-012-0024-x

Keywordsnull sets, Borel measure.

This content is open access.

Keywords
null sets, Borel measure.