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So I’ve taken a probability theory class and I’m studying for a measure theory class next quarter. While I always knew probability theory incorporated a lot of measure theory, it seems to me like it really is just applied measure theory. It incorporates some other notions like Bayesian probability results, but it is otherwise just renaming notions from measure theory and developing special cases of measures and $\sigma-$algebras. Is this an unjustified understanding of probability theory?

Miles Gould
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    Yes, but... Many of topics in probability (those connected with stochastic processes, for example) are motivated by applications (in physics, finance, etc) and do not appear in classical measure theory courses. – kimchi lover Jul 15 '23 at 15:00
  • @kimchilover Yes, of course. I think it may have been the way my professor presented probability theory. It was presented in a very abstract, pure mathematical framework. The entire class was framed around this fairly deep result about sequences of coin flips, and the textbook we used didn’t involve many real life applications. – Miles Gould Jul 15 '23 at 15:04
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    Relevant https://math.stackexchange.com/questions/118221/what-distinguishes-measure-theory-and-probability-theory. – Akira Jul 15 '23 at 15:09
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    I heard it once from Leo Brennan: probability is measure theory with a soul. – Mittens Jul 15 '23 at 18:47
  • As a new PhD student in Statistics years ago I sat in on a grad-level electrical engineering course on information and signal theory. The first day was "Review of basic statistics", so I thought ok, this will be a breeze, but then the professor went through so many concepts that I had never heard before - Lebesgue integration, σ-algebra. That was the first day in school that I felt schooled . Literally that day an operations research prof later described all of stats as just applied optimization. So it's all relative. – prototype Jul 16 '23 at 05:03
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    I believe it's Terence Tao who said (I'm paraphrasing): "Saying probability is the study of measure spaces with total measure 1 is like saying number theory is the study of finite strings of digits." – Carmeister Jul 16 '23 at 06:16
  • @Carmeister Of course Tao said it better, but my first thought on seeing this question was "aren't fields just rings with inverses?" Well yes, but that means a lot more than you might think. – Charles Hudgins Jul 16 '23 at 20:55
  • @Carmeister Sure, but would you not also agree that field theory is just an application of ring theory, as it is just a special (though extremely fruitful) case? – Miles Gould Jul 16 '23 at 21:22

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I guess you can think about it that way if you like, but it's kind of reductive. You might as well also say that all of mathematics is applied set theory, which in turn is applied logic, which in turn is ... applied symbol-pushing?

However, there are some aspects of "measure theory" that are used heavily in probability, but don't arise nearly as much in other applications. Independence is a big one, and more generally, the notion of conditional probability and conditional expectation.

It's also worth noting that historically, the situation is the other way around. Mathematical probability theory is much older, dating at least to Pascal in the 1600s, while the development of measure theory is often credited to Lebesgue starting around 1900. Encyclopedia of Math has Chebyshev developing the concept of a random variable around 1867. It was Kolmogorov in the 1930s who realized that the new theory of abstract measures could be used to axiomatize probability. This approach was so successful that everybody started using it, leading to the modern perspective that makes probability look like a branch of measure theory.

This also explains why probability seems to have "its own words" (random variable, expected value, etc) for measure theory concepts (measurable function, integral, etc): probability had those concepts first, though defined differently, and identifying them with the measure theory concepts came later.

Nate Eldredge
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    I agree that it is reductive, though I don’t think the situation is the same as calling math applied logic, specifically for the historical reason you mentioned. Probability historical preceded measure, though math did not precede logic. It is more like saying analysis is applied topology, which I think is a completely fair characterization. We came to discover one subject as being a special case of a larger subject which is useful in a generalized sense. – Miles Gould Jul 15 '23 at 15:20
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    @MilesGould: It was mostly a joke, but I was just making the point that a field X can have its own identity even if it is defined entirely in terms of concepts from a field Y. – Nate Eldredge Jul 15 '23 at 15:23
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    “applied symbol-pushing” haha that’s a new one – peek-a-boo Jul 15 '23 at 15:27