What is the motivation behind sigma-algebra properties?

Question

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:

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

2. 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.

Answer 1

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.

Answer 2

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.