5 November 2010

The fear of OMEGA

A few weeks ago I finished teaching (yet another time) a sort-of upper division undergraduate probability course. What I want to talk about is the beauty and fear of Ω.

As everybody knows, many undergraduate texts in probability start (pompously so) by putting the subject in its proper basis: A probability space is a triplet (Ω, F, P), where Ω is a set, F is a sigma-algebra of subsets of Ω and P is a countably additive function from F to the nonnegative real numbers such that P(Ω)=1.

And then they go on by giving the reader (only) some trite (silly) examples of probability spaces (such as the set {1,2,3,4,5,6}). After going throuh this rite, the quickly forget Ω.

Poor Ω, you seem to be condemned to death right away, from the start. We talk about you, we make you appear stupid, and then we tell the students: We shall not use this from now on.

What makes things worse is that when we speak of random variables, we immediately tell our students that we shall never write X(ω), but, simply, X. There are, of course, very good reasons for doing so, and, indeed, many times, we need not think of random variables as functions, but, simply, be able to handle probabilities associated with them.

In doing so, we immediately destroy the power of Ω, and tell the student that it's not really there. We condemn it to death. We make students fear of them. Some students graduate, they go to get a Master's, maybe a PhD later, and they reach the professorial levels, all the way having the fear of Ω. So much so, that they often miss a huge part of Probability because they are unwilling to delve into Ω and see that it is there and exists!

I am starting a campaign: Re-introduce Ω and keep it up to the surface, by giving, right from the beginning, meaningful examples where the construction (rather than the axiomatization) of Ω is used.

It took people long time to talk about Probability correctly and now what? Should we pretend we don't know what it is? And keep going on teaching the subject as if it were not understood?

No, I am NOT claiming we should teach it a la Bourbaki. No. I am just saying that, while we do speak of Probability in terms of dice, coins, coincidences, noise, etc., let us not forget that it lives on some Ω which can be used, whenever convenient.


  1. I have a related campaign. If I could go back a few decades, I would rename a measurable function from the probability space as an "observable". I think this fits nicely with intuition, for example that you run an experiment, or you look at the world, and can "observe" various things about it. It would help your campaign, because it emphasises the relationship between $\omega$ and $X(\omega)$.

    Certainly it's better than "random variable". For example, people often seem to find the joint distribution of two random variables to be a murky concept. But the joint distribution of two observables is a very natural idea.

  2. Good idea James. We all dislike the fact that a random variable is, for many, a murky concept, and it is hard to convince the beginner that a random variable has nothing to do with randomness--mathematically speaking.

    I remember, a few years ago, that Mumford (who became a fan of Probability in the second half of his career) proposed a way to aziomatize Probability by taking the concept of a "random variable" as primitive. I don't know how serious the suggestion was.

    Recently, I saw that David Aldous is conducting a one-man campaign in order to retire "dice" as the icon for randomness!



What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant