Reference A convenient category for higher-order probability theory [heunen-kammar-staton-yang-2017]

Introduces quasi-Borel spaces: sets equipped with a class of "admissible random elements" from \RR , closed under precomposition with measurable maps and under countable measurable case-splits. The category is cartesian closed with a commutative probability monad, fixing the failure of cartesian closure of \Meas . Relevant here for the site it implicitly certifies: quasi-Borel spaces are the concrete sheaves on the category of standard Borel spaces equipped with the coverage of countable measurable partitions, so the full topos of sheaves on that site is a natural "measurable-space-like" Grothendieck topos in which QBS embeds. Used downstream in Vákár–Kammar–Staton.

@inproceedings{heunen-kammar-staton-yang-2017, title={A convenient category for higher-order probability theory}, booktitle={Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)}, author={Heunen, Chris and Kammar, Ohad and Staton, Sam and Yang, Hongseok}, year={2017}}