In probability theory, a probability space is a measure space $(X,\mu)$ whose measure $\mu$ is a probability distribution: its integral is $\int_X \mu = 1$ (e.g. Dembo 12, 1.1).
One thinks of the elements $x\in X$ as possible configurations of a system subject to randomness, hence of $X$ as a space of “possible worlds” in the idealized situation under consideration, and for any subset $U \subset X$ one thinks of $\int_U \mu$ as the probability that the system is found in a configuration $x$ which lies in $U$.
Accordingly, a measurable function $f$ on a probability space has the interpretation of a random variable. Its integral $\langle f\rangle \coloneqq \int_X f \cdot\mu$ is its expectation value.
quantum probability space?
The modern formal concept originates around
Surveys and lecture notes include
Last revised on December 11, 2017 at 08:53:08. See the history of this page for a list of all contributions to it.