Measurable Functions and Their Integrals

Abstract

For a given basic measure space, the basic measurable sets and the corresponding step functions form quite limited classes. Many of the sets, and actually most of the functions, which arise in analytical theory and practice will not be in these classes. It is apparent, for example, in the case of the Lebesgue basic measure space on the real line, that the integration of bounded continuous functions over finite intervals is not covered by the theory of integration for step functions. This is, however, as it should be. The general program in the development of the theory of integration should commence with a limited, yet transparent and readily constructed notion of integration such as that given in Chap. 2. The next step is to extend the integral to a wide class of functions, including, hopefully, all those of analytical interest, in such a way as to maintain all such useful properties as those already derived. That this should be possible is not at all obvious, nor does it have even an appearance of intuitive inevitability. That it is possible in fact, in such a relatively unique fashion, is in its way a quite striking phenomenon.