Is probability theory a branch of measure theory?

[Demystifier]

In another math thread
https://www.physicsforums.com/threads/categorizing-math.889809/
several people expressed their opinion that, while statistics is a branch of applied mathematics, the probability theory is pure mathematics and a branch of analysis, or more precisely, a branch of measure theory. Here I would like to discuss in more detail in which sense this statement is true and in which sense it is not.

Let us start from wikipedia
https://en.wikipedia.org/wiki/Probability_theory
which says
"Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion."
From that quote, it seems pretty clear that probability theory is not merely a branch of measure theory.

But why then many mathematicians think it is? I think the main reason is the Kolmogorov axiomatic approach to probability
https://en.wikipedia.org/wiki/Probability_axioms
where probability theory is really presented as a branch of measure theory. For many mathematicians such an axiomatic approach is the only mathematically rigorous way to study probability. Hence, if probability is a branch of mathematics at all, it is a branch of measure theory.

That makes sense from a purely mathematical point of view, but I think such a purely mathematical perspective is a too narrow perspective. From a more philosophical perspective
https://en.wikipedia.org/wiki/Probability_interpretations
the axiomatic approach to probability is just one of many approaches. So from a purely mathematical point of view it is OK to say that probability is a branch of measure theory, but pure mathematicians do not have a monopole on the concept of probability. Probability is used in many other human activities (besides pure mathematics), and in most other uses of probability it does not make much sense to think of it as nothing but a branch of measure theory.

To conclude, measure theory is an important aspect of probability, perhaps even the only mathematically well defined aspect or probability, but still it is not the only relevant aspect of probability. Probability also has something to do with knowledge and randomness, which are important concepts that cannot be reduced to measure theory.

 

[Stephen Tashi]

Probability is used in many other human activities (besides pure mathematics), and in most other uses of probability it does not make much sense to think of it as nothing but a branch of measure theory.

Are you raising a question about psychology - how "to think of it" ?

People will think of words in various ways. How people choose think about "probability" is a subjective question.

(Similarly, how people think about other mathematical concepts, such as "the real numbers" is an issue of psychology.)

Certain ways of thinking may impede one's ability to understand the pure mathematical approach to a topic. That doesn't mean those ways of thinking aren't useful for other purposes.

 

[Demystifier]

Are you raising a question about psychology - how "to think of it" ?

That was not my intention, but I guess my question can be interpreted in that way.

My main motivation was categorization of whole math into branches and sub-branches, and it turned out that probability theory seems to be most problematic. Perhaps it is related to the fact that there is no good mathematical definition of "randomness", so with a desire to well define probability mathematicians translated probability into something which has nothing to do with randomness.

 

[Stephen Tashi]

That was not my intention, but I guess my question can be interpreted in that way.

Do you want it to be interpreted that way? What other interpretation is possible ?

It's would be of some interest to hear various people give testaments on the topic of "How I Think About Probability". Learning how other people think can be of practical use.

 

[Demystifier]

Do you want it to be interpreted that way? What other interpretation is possible ?

See the edited version of my post #3.

It's would be of some interest to hear various people give testaments on the topic of "How I Think About Probability". Learning how other people think can be of practical use.

Yes, that would be great.

 

[Stephen Tashi]

See the edited version of my post #3.

How things ought to be categorized by a individual person is a subjective question.

How things are categorized for some specific purpose is a somewhat objective problem. However, unless you can state a specific purpose for categorizing mathematics (i.e. what specific criteria make one way good and another way bad), people will assume you are discussing how to categorize mathematics in accordance with traditions.

Yes, that would be great.

I'll get around to composing a little essay on that topic - and we'll see if you still think that!

 

[Demystifier]

Pure mathematician like to see probability and statistics as two rather different topics. As a rule, books having both probability and statistics in its title are written for scientists and engineers (who think of probability and statistics as closely related subjects), not for mathematicians. I have found only one exception:
D. Williams - Weighing the Odds: A Course in Probability and Statistics
https://www.amazon.com/dp/052100618X/?tag=pfamazon01-20
which is written for mathematicians. Interestingly, this book has sections entitled
1.3 Probability as Pure Maths
1.4 Probability as Applied Maths
clearly showing that there are two different views of probability.

 

[fresh_42]

Probability also has something to do with knowledge and randomness, which are important concepts that cannot be reduced to measure theory.

But isn't "knowlegde" simply a time dependent additional parameter? What immediately came to my mind were Markov chains, i.e. another analytical approach. And about "randomness" I remember a discussion here on PF on which the participants vehemently insisted that random variables / functions to be "proper" real-valued mathematical objects, that have nothing to do with randomness at all.
In my opinion, the entire question is whether one's approach is historical and application driven (economy, statistics, etc. etc.) or analytical by measure theory. I find it kind of artificial to distinguish between them. In order to cover all aspects of probability theory, you cannot ignore one of them.

My main motivation was categorization of whole math into branches and sub-branches, and it turned out that probability theory seems to be most problematic. Perhaps it is related to the fact that there is no good mathematical definition of "randomness", ...

Categorization always has to lead to compromises and false classifications, depending on whom you ask. E.g. I found the rigorous statement, to think of probabilities as ordinary real-valued objects very pleasant and enlightening. It kind of freed me from some ballast in my thinking.

"...so with a desire to well define probability mathematicians translated probability into something which has nothing to do with randomness."

But isn't this something we do all the time, completely without thinking about it? We translate real world concepts into something which has nothing to do with velocity, forces, mirror symmetries, weight, length and so on. I remember a professor of mine who started his lecture on differential equations by the statement, that there is nothing smooth out there and everything in the real world is discrete. (Please don't discuss this here, I know it's debatable.) And of course has the rest of his lectures mainly been on smooth functions anyway.

So the question about "the mathematical description of randomness as something independent from randomness" is to me the same as to describe symmetries by groups, or curvatures by infinitesimal changes.

I guess in the end, the only difference is, that stochastic is comparably young and simply used by the "wrong" people: economist, social scientists, medicines and so on! And I don't see physics in here, yet. One can well handle wave functions without using the term expectation value. Perhaps this is currently (and slowly) changing as ever more physicists are used to particle physics where you need stochastic concepts to evaluate experiments.

Do you want it to be interpreted that way? What other interpretation is possible ?

Too bad we can't ask Wittgenstein.

 

[Stephen Tashi]

clearly showing that there are two different views of probability.

There are hundreds of different ways to think about "probability" ( or "loyalty" or "justice" etc.) What are you trying to demonstrate? Are you making a point about culture and sociology - i.e. that there are people with academic and technical skills who think about "probability" in certain ways? Are you seeking a scheme of classifying mathematics that is based on such social "norms" ?

 

[pwsnafu]

That makes sense from a purely mathematical point of view, but I think such a purely mathematical perspective is a too narrow perspective. From a more philosophical perspective
https://en.wikipedia.org/wiki/Probability_interpretations
the axiomatic approach to probability is just one of many approaches. So from a purely mathematical point of view it is OK to say that probability is a branch of measure theory, but pure mathematicians do not have a monopole on the concept of probability. Probability is used in many other human activities (besides pure mathematics), and in most other uses of probability it does not make much sense to think of it as nothing but a branch of measure theory..

So how does anything you said change the mathematics? At most it changes how one interprets the results of a calculation using probability. It doesn't change the underlying mathematics. That's like arguing between "e is the logarithm base with derivative equal to 1/x" or "e is continuously compounding interest rate". From a mathematical point of view, e is e. It's not more or less.

 

[FactChecker]

My two cents:
When I think of standard probability theory topics (like Bayes' rule, Stochastic processes, Markov chains, etc.), I rarely have to consider the standard measure theory topics (like measurable subsets, Lebesque integration, etc.). In fact, a standard reference like Feller's "An Introduction to Probability Theory and its Applications" would have very little of the measure theory topics. The converse also holds. So even though the two are closely related, I don't tend to think of them as "branches" of each other.
It probably depends on what criteria one uses to call something a "branch". We could conceivably call everything a branch of either set theory or logic.