ABSTRACT
The theory of the Riemann integration is typically developed in courses in calculus. The Riemann theory is extendable in certain contexts to unbounded functions and/or unbounded intervals, and conventionally if not humorously labelled “improper” integrals. This chapter provides a brief review of the Riemann approach to integration, and discusses an alternative theory of Lebesgue integration. The idea underlying the construction of the Lebesgue integral is also fundamental to a host of generalizations to other measures, and to applications, most notably in probability theory. The Riemann integral is proved to exist for continuous functions on bounded intervals and generalized. The Lebesgue theory preserves virtually all of the familiar properties of the Riemann counterpart, yet it eliminates all of the anomalous properties related to sets of measure 0.
