CHAPTER 8 - COMPARISON OF INTEGRALS

Summary

In preceding chapters the generalized Riemann integral has been presented on its own terms with explicit reference only to the Riemann integral. Nevertheless, the theory of the Lebesgue integral has been in the background at all times. At many points it has given guidance as to what could be proved. At some points the methods of proof were drawn from the Lebesgue theory, too. The existing literature on integration centers on the Lebesgue theory. Thus it is very helpful to know how the generalized Riemann integral ties in with the Lebesgue integral in order to be able to use other books.

Measure and measurable sets as defined in Section 4.6 are identical to Lebesgue measure and Lebesgue measurable sets. The absolutely integrable functions are identical to the Lebesgue integrable functions.

The key to comparing the two senses of measure and measurable sets lies in characterizing measurable sets in a fashion independent of the gauge limit. This is done in Section S8.1 in terms of sets which are the intersection of a sequence of open sets. (They are called G_δ sets.) The crucial element in this discussion is the covering lemma which was given in Section S5.5.

The Lebesgue measurable sets have exactly the same characterization as the one given in Section S8.1. Knowing that the measurable sets are the same in the two theories, one must go on to see that the measures are identical.