[Math] Difference between Real Analysis and Probability Theory

probability

I do not really see a big difference between the two subjects. I was wondering if somebody can explain what the big difference between them is.

Let us compare the superficial differences:

  1. In real analysis our subsets are called "measurable sets", in probability our subsets are called "events". The measure of a set in analysis is called the "measure", while in probability it is called "probability".
  2. In real analysis we deal with "measurable functions", in probability theory we deal with "random variables".
  3. In probability theory random variables induce "distributions", while in real analysis they are more naturally called "push-forwards".
  4. In analysis we "integrate" with respect to the measure, in probability we compute the "expected value".
  5. In analysis we say "almost everywhere" in almost every theorem, and in probability we say "almost surely" in almost every theorem.

There is one major difference:

  • Probability theory assumes that we have a finite measure normalized to be equal to 1.

Other than that last part everything else seems to be essentially the same. It is the "finite measure assumption" which makes probability theory "work".

The only difference that I see is that, analysis is more general than probability theory. In mathematics we often require more generality with a compromise of some of its theorems. Is there something more?

Best Answer

There is a huge difference. The key additions are the concepts of independence (of sigma-fields), conditional independence (given a sigma-field), and conditional expectation/probability (given a sigma-field), which don't play a central (if any) role in Real Analysis. Probability and Statistics without the concept of conditional independence are hardly possible, and definitely boring. In my opinion, Kolmogorovov's "Grundbegriffe der Wahrscheinlichkeitsrechnung" major contribution is the introduction of the general definition of conditional expectation (which depends on the Radon-Nikodym machinery). The importance of this concept in the development of modern Probability and Mathematical Statistics is hard to overstate. Take a look at "Probability with Martingales" by David Williams, and "Theory of Statistics" by Mark J. Schervish.

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