$\Gamma$ is an outer mesure of $X$
\\ 
$\begin{aligned}
 & 1.\Gamma(\phi)=0 
\\ 
 & 2.A\subset B\Rightarrow \Gamma(A) \leq \Gamma(B)
\\
& 3.\Gamma(\displaystyle\bigcup_{n=1}^{\infty}A_n) \leq \displaystyle\sum_{n=1}^{\infty}\Gamma(A_n)
\end{aligned}$


Outer measure


$\Gamma(A)=\inf\displaystyle\sum_{n=1}^{\infty}m(E_n)$
\\
$\begin{aligned}
 & 1.\Gamma\text{ is an outer measure}
\\ 
 & 2.m\text{ is complete additive on }\mathcal{F}\Rightarrow \forall E\in \mathcal{F} (\Gamma(E)=m(E)) 
\end{aligned}$


Outer measure

$Outer measure$

$Theorem$