What is "algebra" in $\sigma$-algebra (or "field" in $\sigma$-field)?
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"?
A field of sets is a family $\mathcal F$ of subsets of a given set $X$ satisfying the axioms:
- $X\in \mathcal F$
- For any $A,B\in \mathcal F$ we have $A\cup B\in \mathcal F$.
- For any $A\in \mathcal F$ we have $X\setminus A\in \mathcal F$.
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).