ABSTRACT
In this chapter, the author knows how to develop and use an integration theory that is based on finite intervals of the form for bounded functions. From our discussions of the Riemann - Stieltjes integral, he knows that the Riemann integral can be interpreted as a Riemann - Stieltjes integral with the integrator given by the identity function. If he finds a way to extend the usual length calculation of an interval to the full sigma -algebra, the author extends the notion of integration as well. He must focuses on developing a theory that can handle integrators which are mappings defined on a full sigma-algebra. It is time to precisely define what the author means by such a mapping. The author discusses the idea that a property holds except on a set of measure zero.
