Studying probability from a non-measure-theoretic text: how much am I missing?

Question

Recently I took a Master's-level class on probability. I was very interested in the material, but the class itself was average. It used a textbook which I didn't care for (by Grimmett and Stirzaker). I've been looking for another text on probability to deepen my knowledge. I am familiar with measure theory, so I am fine with a book that employs it.

I am looking at Feller's book An Introduction to Probability, which seems to be well-regarded. I really like the examples and the way the material is covered. One thing I'm concerned about is that the book does not seem to involve measure theory.

How much of the soul of probability will I be missing if I read a book which covers it from non-measure-theoretic perspective? Will I have to consult another resource if I want to really understand the heart of what's going on?

Perhaps I should use Feller to become acquainted with the basics of probability theory, and supplement it by referring to a more theoretical text for the "real" proofs of certain theorems?

Answer 1

To my mind the use of measure theory in probability is needed to generalise the notion of "probability" to abstract spaces. In some ways, this is quite natural and leads to the use of tools developed in measure theory. For example, we need a broader notion of an integral (say) to cope with the generality (rather than the classical Riemann integral) or a broader and better defined notion of which subsets of the sample set correspond to the intuitive notion of an "event" etc.

This process of abstraction (as is often the case in mathematics) results in a more complicated theory, but provides a consistent framework, where for example, a probability space with a discrete finite sample space, and a probability space with an uncountable sample space can be viewed as particular instances of the same theory.

The question "How much of the soul of probability will I be missing" is more difficult to answer, as in some sense this is subjective. My view is that the soul of probability theory is contained not in the generalisation mentioned above, but rather in the intuitive notion of probability as espoused in Feller (say). However, it is also viable to take the view, that the more abstract/axiomatic version of probability is indeed the "soul" of the subject, and that the abstraction leads to a more coherent and broad view of the subject. The answer here is a matter of taste (I believe).

In short, given that you have already studied measure theory and real analysis to some degree, and you have the intuition gained by reading Feller, I would suggest ploughing through the measure theoretic formulation as you may find that it indeed contains the "soul" of the subject.

A reference that you may be interested in is "An Introduction to Probability and Measure" by J.C Taylor. It is well written, covers the main aspects of the subject (includes a nice interlude on martingales) and has many examples to build intuition.

PS. Another factor that may influence your choice of whether to study probability from a measure theoretic standpoint is that probability theory is an enormous subject, and if you hope to continue beyond the point of a general understanding, then measure theory is a must.