ABSTRACT
Integration is one of the most important and basic concepts of analysis. While the theory of Riemann integration is satisfactory for solving many problems of pure and applied mathematics, there are some serious shortcomings to this theory that make it inadequate in many areas of mathematics and the mathematical sciences. This chapter looks at sequences of measurable functions and their convergence properties. These properties along with the above theorem help us define the Lebesgue integral for certain measurable functions. The Lebesgue integral is defined for simple functions, and its properties are studied on convergent sequences of simple functions. The properties of mean convergence and convergence in measure imply that the absolute values, constant multiples, and finite sums of integrable functions are integrable functions.
