I am learning that a $\sigma$-algebra over a set $S$ is a set of subsets of $S$ satisfying certain properties (see e.g. ProofWiki).
What do the letter "$\sigma$" and word "algebra" refer to or signify?
(I am most familiar with the use of "$\sigma$" as the symbol for standard deviation, but I'm pretty sure that's not how it's used here. I also know of elementary-school algebra and abstract algebra, but I'm not sure what the connection is here.)
$\sigma$ is a Greek expression derived from the German word 'summe' , back in those days Union of sets was understood as addition of sets! So $\sigma$ is used to represent the idea of countable Union. Algebra is all about defining some operations on set and then studying it[defined structure] under the restrictions of those operations. I hope now it'll [ $\sigma$ -algebra] make sense for you!