In a measure theory course we have just learned about $\sigma$-algebras and Borel sets. We defined a $\sigma$-algebra as a collection $S$ of sets which satisfies the following:
- $S$ is closed under complementation of sets.
- $S$ is closed under countable union.
Now, I'm wondering what one such collection might look like or be defined as in $\mathbb{R}^n$ and hopefully someone can help me out here.
This question and its answers helped to ground my understanding a bit more but I'm still looking for more. Any help is appreciated.
One example of a $\sigma$-algebra in $\mathbb{R}^n$ is the Borel $\sigma$-algebra, defined as the smallest $\sigma$-algebra that contains all the open sets in $\mathbb{R}^n$. This is a basic and important example of a $\sigma$-algebra that you should know about. If you have not yet read about how a $\sigma$-algebra is generated from any arbitrary subset of the power set, then do so and come back to this example.
Another closely related example is the $\sigma$-algebra of Lebesgue measurable sets. It is the completion of the Borel $\sigma$-algebra. If you haven't seen these ideas in your course yet, then you will be seeing them quite soon. Complete measures are also quite important.
In the question that you've linked, the two trivial examples of $\sigma$-algebras are also mentioned: namely the power set $\mathcal{P}(\mathbb{R}^n)$ and the collection $\{ \emptyset, \mathbb{R}^n \}$.
Additionally, one small nitpick. You must define a $\sigma$-algebra in a set $X$ to be a collection of sets that satisfies the two conditions you mentioned, and also has $X$ as a member. Don't forget the last condition.