Sigma-algebra

Definition

Let X be some set, and let 2X represent its power set. Then a subset ${\displaystyle \Sigma \subseteq 2^{X}}$ is called a σ-algebra if it satisfies the following three properties:[1][2].

1. X is in ${\displaystyle {\mathcal {S}}}$, and X is considered to be the universal set in the following context.
2. Σ is closed under complementation: If A is in Σ, then so is its complement, X \ A.
3. ${\displaystyle {\mathcal {S}}}$ is closed under countable unions: If A1, A2, A3, ... are in ${\displaystyle {\mathcal {S}}}$, then so is
${\displaystyle A=A_{1}\cup A_{2}\cup A_{3}\cup \ldots =\bigcup _{i=1}^{\infty }A_{i}}$

Lemmas

From 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 ${\displaystyle \emptyset }$ is in ${\displaystyle {\mathcal {S}}}$, since by

• (1) X is in ${\displaystyle {\mathcal {S}}}$
• (2) asserts that its complement, the empty set, is also in ${\displaystyle {\mathcal {S}}}$. Moreover, since ${\displaystyle \{X,\emptyset \}}$ satisfies condition
• (3) as well, it follows that ${\displaystyle \{X,\emptyset \}}$ is the smallest possible σ-algebra on X.
• The largest possible σ-algebra is the power set on X, which contains is ${\displaystyle 2^{n}}$ elements, if X is finite and contains n elementss.

Elements of the ${\displaystyle \sigma }$-algebra are called measurable sets. An ordered pair ${\displaystyle (X,{\mathcal {S}})}$, where X is a set and ${\displaystyle {\mathcal {S}}}$ is a ${\displaystyle \sigma }$-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 ${\displaystyle \sigma }$-algebra to [0, ∞].

A ${\displaystyle \sigma }$-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).

• Explore the measurement problem and explain why a ${\displaystyle \sigma }$-algebra is necessary for the definition of probability distributions!