ABSTRACT
This chapter begins the development of Lebesgue integration, which constitutes Part I of the text. The theory may be seen as arising from the need to overcome some of the shortcomings of the Riemann integral, which is restrictive in both the kind of function that may be integrated and the space over which the integration takes place. These shortcomings make the Riemann integral unsuitable for certain applications, for example those involving random parameters. A further complication with the Riemann theory concerns the integration of a pointwise limit of a sequence of Riemann integrable functions, such limits sometimes failing to be Riemann integrable. The removal of these limitations may be seen as a reason for the wide applicability of the Lebesgue theory.
