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.