# σ-Algebras

Definition (Sigma-Algebra, Measurable Space, Measurable Set)

A

σ-algebraon a set $X$ is a collection $A$ of subsets of $X$ such that

- $X$ belongs to $A$,
- if $A∈A$, then $X∖A∈A$,
- the union of any countable subcollection of $A$ belongs to $A$.
A

measurable spaceis a pair $(X,A)$ consisting of a set $X$ and a σ-algebra $A$ on $X$.

The subsets of $X$ belonging to $A$ are calledmeasurable sets.

Example On every set $X$ we have the σ-algebras ${∅,X}$ and $P(X)$.

If $A$ is

σ-algebraon a set $X$, then:

- $∅$ belongs to $A$,
- if $A,B∈A$, then $B∖A∈A$,
- the intersection of any countable subcollection of $A$ belongs to $A$.

## Generated σ-Algebras

Proposition (Intersection of σ-Algebras)

If ${A_{i}}$ is a family of σ-algebras on a set $X$, then $⋂_{i}A_{i}$ is a σ-algebra on $X$.

Definition (Generated σ-Algebras)

Suppose $E$ is any collection of subsets of a set $X$. The

σ-algebra generated by $E$, denoted by $σ(E)$, is defined to be the intersection of all σ-algebras on $X$ containing $A$.

By the previous proposition, $σ(E)$ is in fact a σ-algebra on $X$.