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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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"Algebra" = Boolean algebra (of sets). I don't know why the term "field" came to be used for this. – GEdgar Dec 27 '12 at 13:20
From tomasz reply, I think "field" stands for "field of subsets" which is usually defined by the three axioms above. Since any field of subsets is a Boolean algebra, this probably results in the term $\sigma$-algebra, where "algebra" stands for "Boolean algebra". I am not sure historically which comes first. – learn_maths Dec 27 '12 at 13:38
But why was the word "field" chosen for that? "Algebra" at least previously existed in connection with Boole's work. – GEdgar Dec 27 '12 at 13:40
That I am not sure. The properties of $\sigma$-fields are not the same as those of commutative division rings in general. – learn_maths Dec 27 '12 at 14:07
up vote 4 down vote accepted

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).

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Thanks, I think I got it now. – learn_maths Dec 27 '12 at 13:30
In what context is algebra not synonymous to field? – BCLC Aug 19 '15 at 7:10
@BCLC: offhand, in universal algebra (there an ``algebra'' just a structure with some operations) or in functional analysis (a Banach algebra is almost never a field). And of course Boolean algebras are not in general called fields, as far as I know (except for fields of sets), much less Heyting algebras and similar objects in algebraic logic. – tomasz Aug 19 '15 at 12:16
@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. – tomasz Aug 19 '15 at 12:55

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