measures 4

For measuring various kinds of figures $C$  in $\Omega=([0,1]\times [0,1])\subset \mathbb{R}^2$ , infinite operations are required;
\[  \sup\cup I^l \subset C\subset \inf\cup I^k  \]
,where $I$s  are squares whose area is precisely known and measurable.

You might think this way is a matter of course.  However, it is not always promised that the infinite operations for subsets in $\Omega$  will bring the desired outcomes.
Then, the following things are put in axioms.

If all $I^j\subset \Omega$ , then $\cup I^j\subset \Omega$  $(i=1,2,\cdots)$ .

In other words,

If $I^j\in \mathcal{F}$ , then $\cup I^j\in\mathcal{F}$   $(i=1,2,\cdots)$ ,
where $\mathcal{F}$  is a family of subsets in $\Omega$ .

It indicates $\cup I^j$  is measurable and $m(\cup I^j)$  exists, including $\pm\infty$ .