Probability and Stochastic Processes

Chapter 4Integration Theory

In the previous chapter, we learned about random variables and their distributions. This distribution completely characterizes a random variable. But, in general, distributions are very complex functions. The human brain cannot comprehend such things easily. So the human brain wants to talk about one typical value. For example, one can give a distribution for the random variable representing players' salaries in the NBA. Here, the variability (probability space) is represented by the specific player chosen. However, probably one is not interested in such a distribution. One simply wants to know what the typical salary is in the NBA. The person probably contemplates a career in sports and wants to find out if as an athlete he should go for basketball or baseball. Thus, he is much better served by comparing only one number corresponding to each of these distributions. Calculating such a number is hard (which number?). In this chapter, we construct a theory which allows us the calculation of any number we want from a given distribution. Paradoxically, to calculate a simple number we need to understand a very complex theory.

4.1 Integral of Measurable Functions

Recall that the random variables are nothing more than measurable functions. Let be a probability space. We wish to define for any measurable function f an integral of f with respect to the measure ...