measures 6

Infinite operations in measurements need the conditions to rulers and things that we want to measure.

At first, rulers $m$  have to satisfy,

(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 $) . 

, and measured things $\mathcal{F}$ have to satisfy,

(1)$\Omega, \phi\in\mathcal{F}$ ,
(2)if $a\in\mathcal{F}$ , $a^c\in\mathcal{F}$
(3)if $a_1,a_2,\cdots\in\mathcal{F}$  , then $(\cup a_i) \in \mathcal{F}$ . 

C. Caratheodory proved these conditions were closely linked.