ABSTRACT
In this chapter we develop the Lebesgue integral and present some of its most important properties. In all that follows μ will be a measure defined on a σ algebra ℱ of subsets of Ω. We always define 0.∞ = 0. This may seem somewhat arbitrary and this is so. However, a little thought will soon demonstrate that this is the right definition for this meaningless expression in the context of measure theory. To see this, consider the zero function defined on ℝ. What do we want the integral of this function to be? Obviously, by an analogy with the Riemann integral, we would want this to equal zero. Formally, it is zero times the length of the set or infinity. The following notation will be used.
