nLab
probability space

Contents

Idea

In probability theory, a probability space is a measure space (X,μ)(X,\mu) whose measure μ\mu is a probability distribution: its integral is Xμ=1\int_X \mu = 1 (e.g. Dembo 12, 1.1).

One thinks of the elements xXx\in X as possible configurations of a system subject to randomness, hence of XX as a space of “possible worlds” in the idealized situation under consideration, and for any subset UXU \subset X one thinks of Uμ\int_U \mu as the probability that the system is found in a configuration xx which lies in UU.

Accordingly, a measurable function ff on a probability space has the interpretation of a random variable. Its integral f Xfμ\langle f\rangle \coloneqq \int_X f \cdot\mu is its expectation value.

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

Revised on April 17, 2015 12:54:12 by Urs Schreiber (195.113.30.252)