measure spaces 2 (set functions)

You might want to know a reason why we need to prepare a set function. 

Saying brief words, because a measure is a set function and a measure space is based on a measure (e.g. a set function). However it might not be easy to understand these relations.

As a set function is the mapping from a set to a real number,

the input values or the arguments of the function are sets.

Therefore, the domain of the set function must be a family of sets.

In an elementary level, a function will be defined on a space (or a universal set), 

after creating the space (e.g. $\mathbb{R}^n$). 

We can not give a value of an element of a space to the set function, and

we must give a subset of a space. 

It is not enough to just only define the ordinary space for a set function.

That is to say, for operating a set function, we have to define a space and subsets in the space.

Measure spaces need a set function and a family of sets in the space.  

Usually, a measure space is written by the triplet $(\Omega, \mathcal{F},m)$ , where 

$\Omega$  is a universal set or a whole space, 

$\mathcal{F}$  is a family of subsets in $\Omega$ , and 

$m$  is a measure (a set function).