In a proof I am reading, the writer takes the sigma-algebra generated from an algebra. Since an algebra is closed under finite unions/intersections and complements I assumed that you just "threw in" all the countable unions of sets in the algebra into the algebra to get the sigma-algebra. Therefore, since the countable union of countable unions is countable, every set in the sigma-algebra should be able to be written as the countable union of sets in the algebra (that is my naive impression at least).
However, the proof is implying that some sets in the sigma-algebra cannot be written as the countable union of sets in the algebra. Could someone explain why or give me an example (maybe the algebra of open sets on the reals has one).
Thank!
Best Answer
Firstly, I must be pedantic and say that the collection of all open sets in $\mathbb R$ is not an algebra (it is not closed under complementation).
Now that the pedantry is over with, let's get to business. Consider the algebra $\mathcal A$ of all finite unions of half-open intervals in $\mathbb R$ of the form $[a,b)$ for some $a<b$ (along with the associated rays $(-\infty, b)$ and $[a,\infty)$). I claim the $\sigma$-algebra this generates is the Borel $\sigma$-algebra on $\mathbb R$ (this isn't too difficult to see). Now let $\mathcal A_\sigma$ denote the collection of all countable unions of elements of $\mathcal A$. This contains all open sets, as each open set is a countable union of open intervals, and open intervals are countable unions of half-open intervals. But I claim that the closed set $$K=\{0\}\cup\{\frac{1}{n}:n\in\mathbb N\}$$ is not in $\mathcal A_\sigma$. To see this, suppose $K=\cup_{n\in\mathbb N}E_n$ where $E_n\in\mathcal A$. Then $0\in E_k$ for some $k$, hence there is some $\varepsilon>0$ such that $[0,\varepsilon)\subset E_k\subset\cup_{n\in\mathbb N}E_n=K$, which is clearly false.
To obtain a $\sigma$-algebra, once you add all countable unions you have to add the complements of all countable unions. Then do this again. And again. And again. And once you've done this countably many times, take the union over everything. Then start over. And once you've done this uncountably many times, you finally have a $\sigma$-algebra. For a better account of this, see the notes at the end of chapter 2 of Folland's Real Analysis.