From the first example in Chapter 1 it has been clear that a function need not be highly regular in order to be integrable. Yet it cannot be wildly irregular. So far we have relied on two kinds of hypotheses to provide sufficient regularity to insure integrability. One type assumes integrability on certain subsets, say all bounded intervals contained in a given unbounded interval. In the other it is the relation of the function to one or more other functions which supplies the appropriate properties. There are two obvious instances of this. One is the relation of |ƒ| to ƒ when ƒ is integrable and integrability of |ƒ| is in question. Convergence theorems are a second. In each of these instances the function under examination gets its regularity “by inheritance,” one might say, from the integrability of other functions.

Since the behavior of Riemann sums provides the criterion for existence or nonexistence of an integral, it has been possible to go very far without identifying the kind of regularity which underlies integration. There are problems which are much easier to solve when it is known just what sort of regularity goes with integrability. The discussion in Section 5.1 goes only as far as identifying the regularity property of absolutely integrable functions.

The appropriate concept is measurability of functions. It is expressed in terms of measurable sets.