What is the motivation behind sigma-algebra properties? Sigma algebras are the fundamental construct that probability theory and Lebesgue integration are based on. I learned a few monographs in probability theory where the term of "sigma-algebra" shows up as a definition, utilitarian, without any discussion around it.
I wonder:


*

*How people came to that construct? I mean how people came to the properties that describe a sigma-algebra.

*Why the properties of sigma-algebra are so unique in conjunction so that they got a special name in math?
There is a similar question on math.stackexchange was raised before but it does not have a clear explanation.
If you know books or internet-resources that shed light on the subject please let me know.

Regarding probability theory, the $\sigma$-algebra definition tries to acknowledge the fact that if you want to be able to ask about the probability of events $A_1, A_2, \ldots$, you might also want to ask about the probability of one of them ($A_k$) not taking place (that is, $A_k^C$); or at least one taking place, which is
$$\bigcup_{k=1}^\infty A_k;$$
or all of them occurring, as in
$$\bigcap_{k=1}^\infty A_k;$$
or none, which is
$$\left(\bigcap_{k=1}^\infty A_k\right)^C=\bigcup_{k=1}^\infty A_k^C;$$
and so on.
Then, when you realize that for this reasonable list of events you want to consider, you just need to ask for certain specific properties that the family of events need to satisfy, you get the definition of a $\sigma$-algebra.
