measure spaces (set functions)

For creating a measure space, some preparations will be required.
These preparations, although using terms are almost known,
if you are not familiar with, may be difficult to understand.


First of all, we shall define a set function.
For an arbitrary set $A$ , if a function gives a real number $x$,
the function is called a set function.

$f: A\rightarrow x\in\mathbb{R}$

(1) for an interval $A=[a,b]\subset \mathbb{R}$ ,
the function $f(A)=|b-a|$ is a set function.

(2) for a finite set $A=\left\{a_1,a_2,\cdots a_n\right\}$ ,
the function $f(A)=n$  which gives a number of elements of $A$
is a set function. This function is called number measure and
usually expressed by $\#(A)$.

Please note that a set function is not a correspondence of values to a value,
and it is a correspondence of a set to a real number.
Namely, a set function is not able to get a real value by giving a value of the set.