The Upper Integral and Some Traditional Approaches to Integration

Abstract

This chapter has a more historical character and its main purpose is to show what is the relation of the matter presented in this book to the integrals of Riemann, of Daniell and the integral generated by a σ-measure. We state the following facts. The Riemann integral is the Lebesgue integral restricted to the class of functions which are continuous almost everywhere. The Daniell integral is nothing else, but the HEM integral in the domain of real functions. Finally, the integral generated by a σ-measure is a HEMPS integral in the domain of real functions. This in particular implies that measure theory is not an adequate tool to construct the Daniell integral, nor the general HEM integral, because both of them are of an essentially wider extent.