σ-algebra

Definition (σ-algebra)

Let $X$ be a set. A $\sigma$ -algebra on $X$ is a family, $\mathcal{F}\subseteq 2^{X}$ , of subsets of $X$ such that

$\emptyset,X\in\mathcal{F}$
$A\in\mathcal{F}\implies A^{c}:=X\backslash A\in\mathcal{F}$
Closure under countable union: $A_{i}\in\mathcal{F}$ , $i\in\mathbb{N}$ (i.e. for any countable collection in $\mathcal{F}$ ) $\implies\bigcup_{i=1}^{\infty}A_{i}\in\mathcal{F}$

A pair $(X,\mathcal{F})$ is a measurable space if $\mathcal{F}$ is a $\sigma$ -algebra on $X$ . Any set $A\in\mathcal{F}$ is called measurable.

Intuition

Think of this as a tool we use in measuring sets’ lengths (based on various notions of length). The two most extreme examples of a $\sigma$ -algebra are the following: $\begin{align*} \mathcal{F}&=\{\emptyset, X\}\\ \mathcal{F}&=P(X) \end{align*}$ With the first satisfying the most basic conditions of a $\sigma$ -algebra and the second being the most extensive version of a $\sigma$ -algebra. We will often be working with versions of $\mathcal{A}$ that are somewhere in-between both examples. ^b0c82d

σ-algebra

Intuition

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