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). 

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.

coffee break 10 ( Happy new year )

New year 2015 has begun.

I would like to continue to pick up a variety of mathematical issues this year.

If you can look at some posts occasionally, I will be so glad.

I want to start from a measure space in next post.

Thanks a lot