The Universal Property of Measure-Theoretic Probability

Building on work of Chen, we give a universal property of the Markov category BorelStoch of standard Borel spaces and Markov kernels between them. To do this, we introduce a new notion of coinflip, or unbiased binary choice, in a Markov category. These are unique if they exist, and automatically preserved by all Markov functors which preserve coproducts. We also provide universal characterizations of various Markov categories of discrete kernels.