measures 10

On an arbitrary space $\Omega$ , if there is an outer measure $m^o:a\subset\Omega

\rightarrow \mathbb{R}$ , the collection of sets
$\mathcal{F}=\left\{a\subset\Omega|\forall e\subset\Omega,m^o(e)=m^o(e\cap a)+m^o(e\cap a^c) \right\}$
could be defined.

It has been proved that for the elements in this collection $\mathcal{F}$ ,
if $a_1,a_2\in\mathcal{F},a_1\cap a_2=\phi$ , then $(a_1\cup a_2)\in\mathcal{F}$ .

Although the proof is not easy, the number of set in the collection can be extended from 2 to $\infty$ . Namely, for $a_1,a_2,\cdots$  and $a_i\cap a_j=\phi$ , then $(\cup a_i)\in \mathcal{F}$ .

The collection $\mathcal{F}$ is called $\sigma$ algebra, and
on $\sigma$ algebra, an outer measure $m^o$  becomes a measure $m$ .