# Is the “field” I learned about in analysis different from the “field” I learned about in econometrics?

Today was my first day of econometrics, and real analysis, and in both courses the professor defined something called a "field". Unfortunately, the field I learned about in real analysis seems completely different from the field I learned about in econometrics.

The field from analysis is a set with addition and multiplication which obeys 11 familiar axioms. I had already been familiar with this definition of a field from linear algebra but here it was again.

But the field from econometrics was completely different! My notes say,

If $S$ is a sample space, a collection of subsets $\mathcal{S}$ of $S$ is called a field if:

1. $S\in\mathcal{S}$
2. Whenever $A\in \mathcal{S}$, $A^C\in \mathcal{S}$
3. Whenever $A$ and $B$ are in $\mathcal{S}$, $A\cup B\in \mathcal{S}$

Is the field that I learned about in econometrics a completely different thing? Or are they somehow related?

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Sounds like a $\sigma$-algebra to me. –  Rahul Sep 7 '12 at 0:01
@RahulNarain $\sigma$-algebras are closed under countable union. –  William Sep 7 '12 at 0:02
@Williamm and Rahul It is sometimes simply called an algebra. –  M Turgeon Sep 7 '12 at 0:05
According to wikipedia, a $\sigma$-algebra is sometimes called a $\sigma$-field. So I guess, some people may call a regular algebra a field. –  William Sep 7 '12 at 0:07
The definition you gave is that of a field of sets (en.wikipedia.org/wiki/Field_of_sets). I suppose in some contexts it might just be called a field. –  Trevor Wilson Sep 7 '12 at 0:13