nLab probability space

Idea

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.

Categories of probability spaces

• 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.

Related concepts

• random variable

• expectation value, moment

• entropy

• quantum probability space

• Prob, category of couplings.

References

The modern formal concept originates around

• Andrey Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Springer Berlin Heidelberg, 1933

Surveys and lecture notes include

• Amir Dembo, Probability theory, 2012 (pdf)

category: probability