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.