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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 ( I suppose in some contexts it might just be called a field. – Trevor Wilson Sep 7 '12 at 0:13
up vote 6 down vote accepted

As Trevor Wilson says in the comments, what you learned about is called a field of sets. A field of sets is in particular a Boolean algebra, and it has two operations (namely intersection and union) which behave formally in some ways like multiplication and addition in a field. For example, intersection distributes over union. But union also distributes over intersection, so this analogy should not be taken too far. And then there is the complement operation too.

There are some other differences, such as the lack of multiplicative inverses and additive inverses. Fields of sets are not actually fields, not even rings, but semirings (or as some call them, rigs).

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Like all Boolean algebras, however, they become rings if we take the ring addition to be symmetric difference rather than union. – Henning Makholm Sep 7 '12 at 16:08

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