Sigma Algebras

A $\sigma$ -algebra allows us to build a further structure on top of our sample space $\Omega$ .

Definition: A collection $\mathcal{F}$ of subsets of $\Omega$ is said to be a $\sigma$ -algebra if $\mathcal{F}$ satisfies the following properties:

$\Omega \in \mathcal{F}$
If $A \in \mathcal{F}$ then $A^c \in \mathcal{F}$ , where $A^c$ is the complement of $A$ relative to $\Omega$ .
If $A = \bigcup_{n=1}^{\infty} A_n$ and $A_n \in \mathcal{F}$ for $n = 1,2,...$ then $A \in \mathcal{F}$ .

Definition: If $\mathcal{F}$ is a $\sigma$ -algebra of $\Omega$ , then $(\Omega,\mathcal{F})$ is said to be a measurable space, and the members of $\mathcal{F}$ are said to be the measurable sets of $(\Omega,\mathcal{F})$ .

Note: We are usually lazy and will just say that $\Omega$ is a measurable space, but we always need to remember that a measurable space is relative to a $\sigma$ -algebra.

Let $\mathcal{F}$ be a $\sigma$ -algebra of a set $\Omega$ , we have the following properties:

Since $\emptyset = \Omega^c$ we have that $\emptyset \in \mathcal{F}$ .
\item If we take $A_{n+1} = A_{n+2} = ... = \emptyset$ we see that finite unions of sets in $\mathcal{F}$ are in $\mathcal{F}$ .
Since
$\begin{align*} \bigcap_{n=1}^{\infty}A_n &= \left( \bigcup_{n = 1}^{\infty} A^c_n \right) ^c \end{align*}$
We have that $\mathcal{F}$ is closed under countable intersections as well.
Since $A - B = B^c \cap A$ we have that $A - B \in \mathcal{F}$ as well.

Now that we have a definition for $\sigma$ -algebras we can move on to discuss Measurable Spaces and Measure Spaces.