# Sigma-algebra

## Definition

editLet *X* be some set, and let 2^{X} represent its power set. Then a subset is called a ** σ-algebra** if it satisfies the following three properties:

^{[1]}

^{[2]}.

*X*is in , and*X*is considered to be the universal set in the following context.- Σ is
*closed under complementation*: If*A*is in Σ, then so is its complement,*X*\*A*. - is
*closed under countable unions*: If*A*_{1},*A*_{2},*A*_{3}, ... are in , then so is

## Lemmas

editFrom these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan's laws).

It also follows that the empty set is in , since by

**(1)***X*is in**(2)**asserts that its complement, the empty set, is also in . Moreover, since satisfies condition**(3)**as well, it follows that is the smallest possible σ-algebra on*X*.- The largest possible σ-algebra is the power set on
*X*, which contains is elements, if*X*is finite and contains*n*elementss.

Elements of the -algebra are called measurable sets. An ordered pair , where *X* is a set and is a -algebra over *X*, is called a **measurable space**. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with the measurable functions as morphisms. Measures are defined as certain types of functions from a -algebra to [0, ∞].

A -algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin's theorem (see Wikipedia:Sigma algebra).

## Learning Task

edit- Explore the measurement problem and explain why a -algebra is necessary for the definition of probability distributions!
- Explain why a discrete probability distribution can use the

## References

edit- ↑ Rudin, Walter (1987).
*Real & Complex Analysis*. McGraw-Hill. ISBN 0-07-054234-1. - ↑ Sigma-algebra. (2017, September 11). In Wikipedia, The Free Encyclopedia. Retrieved 10:03, December 31, 2017, from https://en.wikipedia.org/w/index.php?title=Sigma-algebra&oldid=800163136