I know that $\sigma$ in $\sigma$-algebra stands for the closure under countable union property. What about "algebra"? Surely it cannot algebra over a field or a ring as defined in algebra textbooks. Also why is $\sigma$-algebra also called $\sigma$-field and what is meant by "field"?
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A field of sets is a family $\mathcal F$ of subsets of a given set $X$ satisfying the axioms:
In other words, it's a boolean algebra of sets with the usual operations. Algebra, in this context, is actually synonymous to field. A $\sigma$-field (-algebra) corresponds to a $\sigma$-complete boolean algebra.
Worth mentioning, it actually is quite naturally a ring in the usual algebraic sense (like any boolean algebra). You're right that it can't be a field except the most trivial two-element case (as zero divisors abound).