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 a damn.
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 use. 🏗 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.
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,” June. http://arxiv.org/abs/1906.09507.
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. https://doi.org/10.1063/1.1494475.
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. https://doi.org/10.1214/aop/1176991250.
Diaconis, P., and D. Freedman. 1980a. “De Finetti’s Theorem for Markov Chains.” The Annals of Probability 8 (1): 115–30. https://doi.org/10.1214/aop/1176994828.
———. 1980b. “Finite Exchangeable Sequences.” The Annals of Probability 8 (4): 745–64. https://doi.org/10.1214/aop/1176994663.
Heunen, Chris, Ohad Kammar, Sam Staton, and Hongseok Yang. 2017. “A Convenient Category for Higher-Order Probability Theory,” January. http://arxiv.org/abs/1701.02547.
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. https://doi.org/10.1109/TPAMI.2014.2334607.