Some of the most useful tools of integration theory will be developed in this chapter. They center on two important operations on functions. One is the formation of the absolute value. The other is the limit of a sequence of functions. The behavior of the generalized Riemann integral with respect to these operations exhibits the strength of this integral definition most vividly.

When ƒ is integrable, it is important to be able to tell whether |ƒ| is also integrable. A simple criterion is stated in Section 3.2. This criterion has implications for the calculation of the length of curves. These are also explored in Section 3.2. The proof of the criterion for integrability of |ƒ| is deferred to Section S3.8, since the argument contains a substantial number of technical details.

In the discussion of the integrability of |ƒ| we make use of ΣJ|∫Jƒ| where the sum is taken over the intervals J in a division of I. The link between this sum and a Riemann sum for |ƒ| is given by a technical tool of fundamental importance called Henstock's lemma. Henstock's lemma is presented in Section 3.1, but its proof is deferred to Section S3.7. It should be added that Henstock's lemma plays a role in the proofs of nearly all the deep results about the generalized Riemann integral.