Part 6 of 7
Random Variables and Probability
Measure theory, measurable functions, and integration tools for macro theory.
Consider an experiment that can have several (but finite) outcomes. For example trowing a dice can turn out in getting any number from 1 to 6, or asking someone out can generate an affirmative response, a negative one or perhaps a maybe, or no response at all. A probability function is a function that assigns a value to each possible outcome while satisfying certain rules.
Its clear that when the outcomes are finite, outcomes form a set \(S=\left\lbrace s_{1},\ldots,s_{n}\right\rbrace\), a probability is a list \(\left(\pi_{1},\ldots,\pi_{n}\right)\) such that \(\Pr\left(s_{i}\right)=\pi_{i}\):
- \(\pi_{i}\geq0\) for all \(i\).
- \(\sum\pi_{i}=1\).
It is natural to define other outcomes that are formed by unions of the former ones, like getting an even number when trowing the dice (the union of getting a two a four and a six) or getting a positive answer or a maybe when asking someone out. It is clear that the probability of these new outcomes is defined by the sum of probabilities of the original outcomes used to define them.
Formally we could say that for any set \(A\subseteq S\) we define \(I_{A}=\left\lbrace i|s_{i}\in A\right\rbrace\) and then a function \(\mu:2^{S}\to\left[0,1\right]\) as:
Furthermore we can define the expected value of a real valued function \(f:S\to\mathbb{R}\) as \(E\left[f\right]=\sum\mu\left(\left\lbrace s_{i}\right\rbrace \right)f\left(s_{i}\right)\).
This same discussion can be carried out if the possible outcomes are countably infinite, but it is difficult to generalize it otherwise. The objective now is to study which properties does this kind of function satisfy and how it is generalized to deal with cases where outcomes are arbitrary. The key for this is to realize that a probability is a function that maps sets into the interval \(\left[0,1\right]\), hence the study of functions that map sets into non-negative numbers will provide the necessary theory, these functions are called measures, for obvious reasons.
The following sections draw on the short exposition of measure theory contained in Chapter 7 of Stokey et al. (1989) and complements it with portions of Kolmogorov and Fomin (2012) (chapters 7 to 10). Both these references are introductory although they present all the relevant results. All the material is also covered in a more advanced manner in Kolmogorov and Fomin (1999).
The aim of the course is not to dwell on the mathematical details of the theory but rather to present the most useful results for applications in economic theory. Because of this, many of the proofs will be omitted, and we will only include those that are especially instructive for how the theory is developed. Kolmogorov and Fomin (2012) is a good source for detailed (and easy to understand) proofs.