25. Lebesgue measure. So far, we have mostly been dealing, directly or indirectly, with what used to be called differential calculus, and was clearly separated from integral calculus. The word “integral” has generally been used in two different senses. One is the process of undoing differentiation to find what have been variously described as antiderivatives, primitives, or indefinite integrals. This is the technique of finding explicit formulas for antiderivatives, an art which has lost much of its importance now that there are not only extensive tables of antiderivatives, but also computer programs that can find antiderivatives faster than people can find them by hand. The other meaning of “integral” is what we picture informally as the area between a curve and a coordinate axis, and more formally as the limit of sums that approximate this area. Of course, these two notions are connected by what is (or at least used to be) called the fundamental theorem of the calculus. This chapter is about the second meaning of “integral.” We shall deal only with real-valued functions on subsets of the real line.

The simplest function we can consider, besides a constant function, is a function whose range consists of just two values, say 0 and 1.