Lebesgue Measure

Abstract

In the previous chapter we discussed differentiation in several variable calculus. It is now only natural to consider the question of integration in several variable calculus. The general question we wish to answer is this: given some subset $\Omega$ of ${{\textbf{R}}}^n$, and some real-valued function $f:\Omega \rightarrow {{\textbf{R}}}$, is it possible to integrate f on $\Omega$ to obtain some number $\int _\Omega f$? (It is possible to consider other types of functions, such as complex-valued or vector-valued functions, but this turns out not to be too difficult once one knows how to integrate real-valued functions, since one can integrate a complex or vector-valued function, by integrating each real-valued component of that function separately.)