Intuition/example on the smallest $\sigma$-algebra I have recently begun to learn about $\sigma$-algebra. Here, the set $\mathcal{F}\in P(\Omega)$ (where $P(\Omega)$ denotes the power set of $\Omega$) is a $\sigma$-algebra if $\mathcal{F}$ has the following properties:
\begin{align}
(1):& \ \ \emptyset\in\mathcal{F} \\
(2):& \ \ A\in\mathcal{F}\implies A^c\in\mathcal{F} \\
(3):& \ \ A_1,A_2,...\in\mathcal{F}\implies \bigcup_{i=1}^\infty A_i\in\mathcal{F}
\end{align}
I am struggling to understand the idea of the smallest $\sigma$-algebra. I have read that given a family $U$ of subsets of $\Omega$, the smallest $\sigma$-algebra is the intersection of all $\sigma$-algebras containing $U$. To help my understanding, I considered an example.
Let $\Omega=[0,1]$ and $U=\{[0,0.2],[0.8,1]\}$. Then the smallest $\sigma$-algebra, denoted by $\sigma_U$, is
$$\sigma_U=\{\emptyset\ ,[0,0.2],[0.8,1]\}.$$
Is this correct? I am quite lost.
 A: I'll try to explain this from probability theory perspective.

*

*Maybe someone will correct me here

In probability of discrete sample spaces we denote by $\Omega$ the sample space. For example by speaking of one dice rolling the sample space can be the possible results: $\Omega = {1,2,3,4,5,6}$ and then we can simply take $2^\Omega = \{ \emptyset,\{1\},\{2\},..,\{1,2\},\{1,3\},...,\{1,2,3,4,5,6\} \}$ to be every possible event.
But, when there is a need to speak of not countable sample spaces (which is the need for example when you investigate probability of events that occur on the interval $(0,1)$) we cant take $2^\Omega$ (There are going to be sets like Vitally sets that will cause problems)
Hence, we need a subset of $2^\Omega$ that maintains the important properties: empty event, complement of an event and union of events.
Now, the concept of minimal $\sigma$-algebra is important because intersection of $\sigma$-algebras is also $\sigma$-algebra and we can prevent ourselves from including problematic sets as Vitally sets and take for example the minimal sigma algebra that includes all the open subsets of $(0,1)$ and that is good enough for many applications.
(Of course we still have to prove that each set of this sigma algebra can have probability, but this can be done by Lebesgue measure).
