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.

The motivation for $\sigma$- algebras is to define a family of sets to serve as the domain for a measure $\mu$. It this sense it is clear that the set itself should be measurable, and the empty set should be measurable. Also, if we know the measure of X and the measure of $A \subseteq X$, then $X\setminus A$ should have measure $\mu(X) - \mu(A)$. Furthermore, one would want to be able to approximate the measure of a set by others i.e. if $\mu(A_i)$ is known for every $i$ in some countable index set, then the measure of $\cup A_i$ should exist and be known in some sense or another.
On the other hand one wants to exclude paradoxical examples like Banach-Tarski, so one cannot have a every subset of $\mathbb{R}$ be measurable. For exmaple, if the index set before were allowed to be uncountable, then every subset of $\mathbb{R}$ would be measurable and the theory would not behave as one would like.
Also, note that $\sigma$- algebras are not the only widely used domain of measures. $\sigma$ -rings and Dynkin-Systems are also suitable - and they have similar properties. So I'm not sure if the properties of $\sigma$-algebras are really that special.
