ABSTRACT
In this chapter, the authors begin the task of describing the integral for functions of n variables. They define integrals in R, prove some basic properties of integrals, and demonstrate that continuous functions are integrable. The authors discuss the theory of integration of functions of one variable. The theory is built on two basic principles. One is that there is an obvious definition for the integral of a step function; the other is that if f ≥ g, then the integral of f is greater than or equal to the integral of g. The authors provide the assertion that a continuous function on a rectangle is integrable. For the integrals of a function over a set which is not a rectangle, however, they need to know that certain discontinuous functions are integrable. The authors describe a result of this sort, using the notion of content zero.
