On the generalized Riemann integral defined by means of special partitions (Q1123278)

Extending the theory of the generalized Riemann integral to higher dimensions is a hard problem that has been the object of much research in recent years. The most successful such extension is due to the present author [Trans. Am. Math. Soc. 295, 665-685 (1986; Zbl 0596.26007)], call it the p-integral. The definition is rather complex and recently a much simpler definition has been given by \textit{A. M'Khalfi} [Bull. Soc. Math. Belg., Sér. B 40, No.1, 111-130 (1988; Zbl 0658.26008)], call this the s-integral. In one dimension both these integrals are the classical Perron integral. Unfortunately in dimensions higher than two the definition of the s-integral appears to be flawed. The present note shows that while, in two dimensions, the s-integral is a proper extension of the p-integral, it lacks some very basic properties. In particular, the indefinite s-integral is neither continuous nor bounded as a function of intervals. As the author remarks, it would appear that ``the added generality of the s-integral is of little value and that the technical complexity of the p-integral may be unavoidable for obtaining an integral with desirable properties''.