The most popular axiomatic foundation for classical commutative probability, i.e. my bread and butter.
I learned about sigma algebras and probability spaces and all the scaffolding of
modern probability in the context of financial mathematics, where
the emphasis is on proving things about certain pathologies of probability
which obtain in certain pathological limits of uncountable
what-have-yous, in order to demonstrate that you are clever
enough to get good grades in order that you can ignore it
for the rest of your career as a financial trader trading in discrete time where
none of those theorems matter.
BUT! There is interesting stuff here. Not just pathologies of measure theory,
but also the interaction between measure theory and the Kolmogorov axioms that
give us actual probability, and there are real problems that can arise in actual
They do not often explain why we bother with this construction here, but there
are practical consequences if we ignore it.
See Djalil Chafaï’s explanation
for a couple of great examples.
Some subtle interactions arise in probability metrics.
Interesting mental exercises here:
conditionalising corresponds with differentiation
Ben Green points out.
Hopefully one day I will have time to return to this with some more profound
insights, but… later.
Campbell, Trevor, Saifuddin Syed, Chiao-Yu Yang, Michael I. Jordan, and Tamara Broderick. 2019. “Local Exchangeability.” arXiv:1906.09507 [Math, Stat]
Caves, Carlton M., Christopher A. Fuchs, and Rüdiger Schack. 2002. “Unknown Quantum States: The Quantum de Finetti Representation.” Journal of Mathematical Physics
43 (9): 4537–59.
Cover, Thomas M., Péter Gács, and Robert M. Gray. 1989. “Kolmogorov’s Contributions to Information Theory and Algorithmic Complexity.” The Annals of Probability
17 (3): 840–65.
Diaconis, P., and D. Freedman. 1980a. “De Finetti’s Theorem for Markov Chains.” The Annals of Probability
8 (1): 115–30.
———. 1980b. “Finite Exchangeable Sequences.” The Annals of Probability
8 (4): 745–64.
Heunen, Chris, Ohad Kammar, Sam Staton, and Hongseok Yang. 2017. “A Convenient Category for Higher-Order Probability Theory.” arXiv:1701.02547 [Cs, Math]
Orbanz, P., and D. M. Roy. 2015. “Bayesian Models of Graphs, Arrays and Other Exchangeable Random Structures.” IEEE Transactions on Pattern Analysis and Machine Intelligence
37 (2): 437–61.
Radul, Alexey, and Boris Alexeev. 2020. “The Base Measure Problem and Its Solution.” arXiv:2010.09647 [Cs]