σ-Algebras

Definition (Sigma-Algebra)

A σ-algebra on a nonempty set $X$ is a collection $\mathcal{A}$ of subsets of $X$ that is

• nonempty,
• closed under complements (if $A\in \mathcal{A}$, then $X\setminus A\in \mathcal{A}$), and
• closed under countable unions (if $\mathcal{C}\subset \mathcal{A}$ is countable, then $\cup \mathcal{C}\in \mathcal{A}$).

Definition (Measurable Space, Measurable Set)

A measurable space is a pair $(X,\mathcal{A})$ consisting of a set $X$ and a σ-algebra $\mathcal{A}$ on $X$.
The subsets of $X$ belonging to $\mathcal{A}$ are called measurable sets.

Example   On any set $X$ we have the σ-algebras $\{\varnothing ,X\}$ and $\mathcal{P}(X)$.

If $\mathcal{A}$ is σ-algebra on a set $X$, then:

• $\varnothing$ and $X$ belong to $\mathcal{A}$,
• if $A,B\in \mathcal{A}$, then $B\setminus A\in \mathcal{A}$,
• the intersection of any countable subcollection of $\mathcal{A}$ belongs to $\mathcal{A}$.

Generated σ-Algebras

Proposition (Intersection of σ-Algebras)

If $\{{\mathcal{A}}_i\}$ is a family of σ-algebras on a set $X$, then ${\bigcap }_i{\mathcal{A}}_i$ is a σ-algebra on $X$.

Definition (Generated σ-Algebras)

Suppose $\mathcal{E}$ is any collection of subsets of a set $X$. The σ-algebra generated by $\mathcal{E}$, denoted by $\sigma (\mathcal{E})$, is defined to be the intersection of all σ-algebras on $X$ containing $\mathcal{A}$.

By the previous proposition, $\sigma (\mathcal{E})$ is in fact a σ-algebra on $X$.

Products of σ-Algebras