measures 7

The properties of a mesure $m$  are, for a $\mathcal{F}$  of $\Omega$ ,

(1) for any $a\in\mathcal{F}$ ,  $m(a)\geq 0$ ,

(2) $m(\phi)=0$ ,

(3) if $a_1,a_2,\cdots\in\mathcal{F}$  ,  $a_i\cap a_j=\phi (i\ne j)$ , then $m(\cup a_i)=\sum m(a_i)$  ($i,j=1,2,\cdots $)  . 

Here, we will weaken the condition (3) to 

(3'-1) if $a_1\subset a_2 (\in \mathcal{F})$ , then $m(a_1)\leq m(a_2)$ , 
(3'-2) if $a_1,a_2,\cdots\in\mathcal{F}$  ,  $a_i\cap a_j=\phi (i\ne j)$ , then $m(\cup a_i)\leq \sum m(a_i)$  ($i,j=1,2,\cdots $)  . 

This means that we will accept to measure a set to large. 
(This means to allow measurement of a given set by approximating it from the outside of its perimeter.)
Such a ruler is called an outer measure $m^o$. 

An outer measure $m^o$  is more natural than the preceding measure $m$  satisfying (3).  

Using the outer measure $m^o$ , we will define the family of sets such that 

$\left\{ a\subset\Omega | \mbox{any } e\subset\Omega, m^o(e)=m^o(e\cap a)+m^o(e\cap a^c)  \right\}$ , 

where $a^c$  is the complementary set of $a$  in $\Omega$. 

C.Caratheodory proved these sets were measurable. Hence, it is called Lebesgue measurable sets.