In another math thread(adsbygoogle = window.adsbygoogle || []).push({});

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 theoryis 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 isnotmerely 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.

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# I Is probability theory a branch of measure theory?

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