9
$\begingroup$

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"?

  • 1
    That I am not sure. The properties of $\sigma$-fields are not the same as those of commutative division rings in general.2012-12-27

1 Answers 1

5

A field of sets is a family $\mathcal F$ of subsets of a given set $X$ satisfying the axioms:

  1. $X\in \mathcal F$
  2. For any $A,B\in \mathcal F$ we have $A\cup B\in \mathcal F$.
  3. 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).

  • 2
    @BCLC: I've missed the obvious: in algebra, an ($R$-)algebra is also often not a field. In fact, there are far fewer contexts when the two *are* synonymous.2015-08-19