Understanding Borel Sets- 2 What would be an example of a $\sigma$-algebra on some sample space $\Omega$ that is not the smallest $\sigma$-algebra on that space?
This is an extension of the question at Understanding Borel sets. I would've posted in the comments but I don't have the rep to do so.
 A: For any space $X$, $\{\emptyset, X\}$ will be the smallest sigma algeba on $X$. So choose any other non-trivial sigma algebra, and it's not the smallest. For instance, the Borel sigma algebra on $\mathbb{R}$ is not the smallest sigma algebra on $\mathbb{R}$.
Your question seems to be asking what is meant by the smallest sigma algebra containing some specified collection of sets, though. Say we want a sigma algebra $\mathcal{F}$ on $X$ that contains all the sets in some collection $\mathcal{A}$ of subsets of $X$. Given any two sigma algebras $\mathcal{F}_1$ and $\mathcal{F}_2$ on $X$ such that $\mathcal{A}\subseteq \mathcal{F}_1$ and $\mathcal{A}\subseteq \mathcal{F}_2$, $\mathcal{F}_1\cap \mathcal{F}_2$ is also a sigma algebra on $X$ that contains $\mathcal{A}$ (check this!). So 
$$\mathcal{F}:=\bigcap_{\text{sigma algebras } \mathcal{G}\supseteq\mathcal{A}} \mathcal{G}$$
is a sigma algebra on $X$ containing $\mathcal{A}$, and it is the smallest such. Why the smallest? Well if $\mathcal{F}'$ is any other such sigma algebra, it is one of the sigma algebras in the intersection defining $\mathcal{F}$, so $\mathcal{F}\subseteq \mathcal{F}'$.
As an exercise, you can convince yourself that the smallest sigma algebra on $X=[0,1]$ containing $[0,\frac{1}{2}]$ is $\mathcal{F}=\{\emptyset,[0,1],[0,\frac{1}{2}],(\frac{1}{2},1]\}$.
A: Start with a set $X \subset \mathbb{R}$ that is not a Borel set, and consider the $\sigma$-algebra it generates. The result is (by definition) a $\sigma$-algebra, but because it contains a set that is not Borel it cannot be the smallest.
