measures 3

There is an arbitrary figure $C$ in $\Omega=( \left[0,1\right]\times\left[0,1\right] )\subset \mathbb{R}^2$ . Then, we want to measure the area of $C$ . If the set function $m$  is a measure of the area in $\Omega$ ,
\[  m : C\in\Omega\rightarrow x\in\mathbb{R}, \]
it is clear that $m(C)\leq 1$ .

However, as we want to measure more precisely any figures $C$, we keep making many various kinds of  small squares $I^k$ whose horizontal size is $c_h$  and vertical size is $c_v$ . Then,  $m(I^k)=c_h^k\times c_h^k,\quad (0\lt c_h,c_v\lt 1,k=1,2,\cdots)$ .

Covering $C$  by some $I^k$ , we can put $C\subset \cup I^k$ and covering some $I^l$  by $C$, we can put $\cup I^l\subset C$ . Hence,
\[  \cup I^l\subset C\subset \cup I^k  \]

Reducing the size of squares and increasing the number of squares,
\[  \sup\cup I^l\subset C\subset \inf\cup I^k  \]
Please note that $\sup\cup I^l$  and $\inf\cup I^k$  must be measurable.

If $l,k\rightarrow \infty$  and
\[  m(\sup\cup I^l)=m(\inf\cup I^k)=m^* , \]
then we will define $m(C)=m^*$ .

You will remember the theorem of Darboux in integral.