# nLab probability space

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## 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.

## 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)