measures 2

Let consider a square of the vertical and horizontal size 1.
Although it is the very small square, if the elements of the square are the combinations of two real numbers $(c_1,c_2)$ , the number of the elements is $\aleph$ ,
and many subsets are included in the square.
\[  \Omega=\left\{ (c_1,c_2)\in \mathbb{R}^2 | 0\leq c_1,c_2\leq 1 \right\}  \]

A simplest subset in $\Omega$  is a arbitrary figure. We want to measure the area of the subset.
Therefore, making many small quadrangles whose area is trivial, we will try to cover the figure with them.

If the figure is also a quadrangle, the figure will be precisely covered with some small quadrangles.  Then, the area of the figure is measured by calculating the small using quadrangles.

However, we can not cover the figure whose shape is curves with any ordinary quadrangles.
Shortage or surplus occurs at the corner.

Infinite operations will be required.

measures

A measure $m$  is a set function which maps a subset in the measure spaces to a real number. 
\[ m: a\in\mathcal{F}\rightarrow m(a)=x\in\mathbb{R} \]
, where $\mathcal{F}$  is a family of subsets of the measure space. 

What conditions are needed for a measure $m$ as a set function?

On standard definitions (or finally reached definitions), it is as follows;

(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 $) 

You may think it is very natural. Perhaps it is fairly accurate in terms of  finite operations or various sets having a number of good shapes. 

However, there are very strange sets (or subsets) in a measure space.  

Therefore, these conditions are closely related with the ones which the families $\mathcal{F}$ of subsets satisfy.