Understanding $\sigma$-algebra and power set Given the set $S =\{1, 2, 3\}$ and the fact that $\sigma$-algebra is generated by its single subset $A = \{1\} \subset S$ and is denoted by $\sigma(\{1\})$ or by $\sigma(A)$. I don't really understand how $\sigma(A)$ is constructed as well as whether the power set of $S$, $\mathcal{F}(S) = 2^S$, is a $\sigma$-algebra?
I have tried to check the $\sigma$-algebra $\mathcal{F}$ properties such as:
$$\emptyset \in \mathcal{F}$$ 
$$A \in \mathcal{F} \Rightarrow A^c \in \mathcal{F}$$  
$$ A_1,A_2, \dots, A_n, \dots \in \mathcal{F} \Rightarrow \bigcup\limits_{k=1}^{\infty} A_k \in \mathcal{F} $$
Therefore, I assumed that $\sigma(\{1\})$ satisfies the properties and therefore may be called $\sigma(A)$. In regard to the power set, I assumed that a legit power set is in the form of $\mathcal{F}(S) = 2^\Omega$ so the question is whether $S=\Omega$? Here I'm a bit confused.
 A: The $\sigma$-algebra $\sigma(\{1\})$ on $S = \{1,2,3\}$is the least subset of $\mathscr{P}(\{1,2,3\})$ which 


*

*has $\emptyset$ as a member,

*is closed under $S$-complements and countable unions, and

*has $\{1\}$ as a member.


It's easy to check that the following collection meets those three conditions:
$$
\mathscr{S} = \{\emptyset, \{1\}, \{2,3\}, \{1,2,3\} \} \text{.}
$$
So $\sigma(\{1\}) \subseteq \mathscr{S}$. But clearly every $X \in \mathscr{S}$ is also a member of $\sigma(\{1\})$, so $\sigma(\{1\}) = \mathscr{S}$.
Because for example $\{2\} \notin \sigma(\{1\})$, $\sigma(\{1\}) \neq \mathscr{P}(\{1,2,3\})$.
It's trivial (yes?) to confirm that for any set $X$, $\mathscr{P}(X)$ is a $\sigma$-algebra: $\emptyset$ is a member; it's closed under complements relative to $X$, and closed under countable unions (the union of any family of subsets of $X$ is a subset of $X$).
A: The power set of $S$ is indeed a $\sigma$-algebra containing $A$. It is the largest such $\sigma$-algebra.
But the $\sigma$-algebra generated by $A$ is the smallest $\sigma$-algebra containing $A$ (in particular, it is the intersection of all such $\sigma$-algebras).
In this case, they are not equal, since, for example, $\{2\}$ lies in the power set of $\{1,2,3\}$ but not in $\sigma(\{1\})$.
