Lebesgue Integration

Abstract

At the beginning of Chapter 2 I attempted to motivate the introduction of measures by indicating how important the role measures play in Lebesgue’s program for constructing integrals of functions. Now that we know that measures exists, in this chapter I will carry out his construction. Once I have introduced his integral, I will prove a few of the properties for which it is famous. In particular, I will show that it is remarkably continuous as a function of its integrand. Finally, in the last section, I will prove a beautiful theorem, again due to Lebesgue, which can be viewed as an extension of the Fundamental Theorem of Calculus to the setting of Lebesgue’s theory of integration.