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:
- $S\in\mathcal{S}$
- Whenever $A\in \mathcal{S}$, $A^C\in \mathcal{S}$
- 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?