measures 5

At first, infinite operations require that a union of sets is in the $\sigma$  algebra.
That is to say,
if all $I^j\in \mathcal{F}$ , then $\cup I^j\in\mathcal{F}$ .

In addition, as we want to measure the area of a figure $C$ , $m(\cup I^j)$  must exist.
Hence, the following thing will be put in axioms.

If all $I^j\in \mathcal{F}$ , and any $I^k\cap I^l=\phi$ , then $m(\cup I^j)=\sum m(I^j)$ .

It means, for example, for any $I^k\cap I^l=\phi$ ,
if $m(I^1)=1$ ,
$m(I^2)=0.4$ ,
$m(I^3)=0.01$ ,
$m(I^4)=0.004$ ,
$m(I^5)=0.0002$ ,
$m(I^6)=0.00001$
$\cdots$ , $\cdots$ , $\cdots$　
, then $m(\cup I^j)=\sum m(I^j)=\sqrt{2}$ .

This proposition shows a precise and adequate measurement for infinite operations.

If $m(\cup I^k)\leq m(C)\leq m(\cup I^l)$  and $m(\cup I^k)=m(\cup I^l), \mbox{ when } k,l\rightarrow \infty $ ,
we will get $m(\cup I^k)=m(\cup I^l)=m(C)$ .