ABSTRACT
This chapter presents a study of a finite Borel measure on ℝn and derives the implied properties on an associated function F that generalize "increasing and right continuous" in the 1-dimensional case. It then generalizes this development to general Borel measures, and shows that any function with these properties induces a Borel measure. In this 1-dimensional case, unbounded intervals were manageable in the sense that there were only two types, and such an intervals either had finite measure or not. For Lebesgue and general measures, the process is more subtle. The approach is axiomatic in that one explicitly defines the value of the integral for certain special functions, and then proceeds to prove that this definition extends to a wide class of functions of interest. One also defines the integral to have an important property with respect to this special class of functions, again with the goal to prove that this property extends to the wider class.
