In probability theory, a probability space is a measure space whose measure is a probability distribution: its integral is (e.g. Dembo 12, 1.1).
One thinks of the elements as possible configurations of a system subject to randomness, hence of as a space of “possible worlds” in the idealized situation under consideration, and for any subset one thinks of as the probability that the system is found in a configuration which lies in .
Accordingly, a measurable function on a probability space has the interpretation of a random variable. Its integral is its expectation value.
The category Prob has probability spaces as objects, and measure-preserving maps as morphisms;
The category of couplings has probability spaces as objects and transport plans as morphisms, or equivalently, equivalence classes of measure-preserving Markov kernels.
The modern formal concept originates around
Surveys and lecture notes include
Last revised on March 17, 2024 at 22:43:58. See the history of this page for a list of all contributions to it.