2Measurable functions and random variables

II Probability and Measure

2 Measurable functions and random variables

We’ve had enough of measurable sets. As in most of mathematics, not only

should we talk about objects, but also maps between objects. Here we want to

talk about maps between measure spaces, known as measurable functions. In

the case of a probability space, a measurable function is a random variable!

In this chapter, we are going to start by defining a measurable function and

investigate some of its basic properties. In particular, we are going to prove the

monotone class theorem, which is the analogue of Dynkin’s lemma for measurable

functions. Afterwards, we turn to the probabilistic aspects, and see how we can

make sense of the independence of random variables. Finally, we are going to

consider different notions of “convergence” of functions.