Symmetric integrals do not have the Marcinkiewicz property (Q1374526)

A theorem of Marcinkiewicz asserts that the classical Perron integrability of a function can be deduced from the existence of a single pair of continuous minor and major functions. The authors construct a measurable function \(f\) that is Denjoy-Khintchin integrable on an interval, but not integrable in either symmetric Perron or approximate symmetric Perron sense. On the other hand, this function possesses a pair of continuous major and minor functions in both senses. Hence, both types of Perron integral fail to have the Marcinkiewicz property. Some other Perron-type integrals which have, or which fail to have this property, are well known. [cf., e.g., \textit{P. S. Bullen} and \textit{R. Výborný}, Can. Math. Bull. 34, No. 2, 165-174 (1991; Zbl 0739.26003)].