# What distinguishes measure theory and probability theory?

It is clear that the theory of probability works primarily with limited measures on measurable spaces. On the other hand, there is a folklore that says that what distinguishes measure theory and probability theory is the conditional probability and conditional expectation.

But conditional probabilities and conditional expectations are derived from the Radon–Nikodym theorem and a measure with respect to another measure. And the Radon–Nikodym theorem is a typical result of measure theory.

Question 1: So we could see the theory of probability as a subdiscipline of measure theory?

Question 2: Would be possible to give another basis for the theory of probability rather than through measure theory?

• Measure theory allows unbounded measures. There exists a uniform measure on the integers, for example, but there is no uniform probability. Commented Mar 9, 2012 at 14:06
• There is a discussion of what distinguishes probability theory from measure theory in section 10.2 of Loève's book Probability Theory I.
– t.b.
Commented Mar 9, 2012 at 15:03
• Saying that probability is a subdiscipline of measure theory is like saying that mathematics is subdisciple of logic.
– Dirk
Commented Mar 9, 2012 at 16:40
• Measure theory is a tool people use, probability theory is a field of mathematics. Commented Mar 9, 2012 at 16:56
• "Probability theory is measure theory with soul". Commented Mar 9, 2012 at 17:00

I would say that conditioning and independence is something that disctinct, expectation is used a lot in the measure theory as well, by the name of the Lebesgue integral.

The point is that the probability as a science before was maybe even more closer to physics than to math by being based on experiments. It became the classical Probability Theory (PT) when it was axiomatized the first half of XX century by the means of Measure Theory (MT). So MT is clearly a mathematical basis for the classical PT and in that sense you can consider PT to be a subdiscipline of MT.

There are two moments to mention, though.

1. There is an algebraic approach to probability which starts with algebras of random variables and defines a linear functional on such algebras - which is an expectation. Shall we say that the Probability Theory is a subdiscipline of Abstract Algebra?

2. In both cases - you start with something empirical: probability, random variables etc. You wish them to satisfy some kind of properties and by this you bring a particular structure: either a measure-theoretical, or an algebraic. However, there is an additional meaning of the results that you obtain. For example, the Law of Large Numbers and Central Limit Theorem are obtained by using pure measure-theoretical methods. But these results are very important exactly for Probability Theory. The interpretation of MT via probabilistic ideas provides you additional intuition about "how should it be" and help to understand "what does it mean".

That is completely an opinion which I've chosen for myself. Hope that it helps.

• That's really interesting. @Ilya have you got any good reference about this algebraic presentation of Probability Theory? Commented Mar 9, 2012 at 14:25
• @ineff: I didn't work with it - just read this note in Terence Tao's blog. That's quite enough to have a first impression
– SBF
Commented Mar 9, 2012 at 14:30
• @Ilya. I appreciate your response. The algebra that differs from the analysis (and particularly the theory of measure) is the axiom of the supreme. Although I am not sure I believe this can be an algebraic basis for the Theory of Probability. But it's like I said. I'm not sure. Commented Mar 9, 2012 at 15:53
• At the moment I'm reading theTerence Tao blog post that you've indicated. I'm thinking interessande. But English is not my native language so it takes me to read. Thank you. Commented Mar 9, 2012 at 17:24
• This other note (linked from the note already posted) gives a lot of discussion about the difference in philosophy between probability and measure theory. Commented Nov 3, 2012 at 10:10

Salomon Bochner's words in the introduction to his article entitled 'Stochastic Processes' published in 1947 in volume 48 of the Annals of Matematics:

Measure theory for its own sake is based on the fundamental addition rule for measures. Probability theory supplements that with the multiplication rule which describes independence; and things are already looking up. But what really enriches and enlivens things is that we deal with lots of $$\sigma$$-algebras, not just the one $$\sigma$$-algebra which is the concern of measure theory.