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.
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.