Let $C$ be any collection of subsets of a set $X$. Let $A(C)$ the algebra generated by $C$. Prove that the $\sigma$-algebra generated by $C$ and $A(C)$ are the same.

Here is my idea of a proof,

Proof: I would like to show that one is a subset of another and vice-versa.

Let $\sigma(C)$ be the $\sigma$-algebra generated by $C$ and $\sigma [A(C)]$ be the $\sigma$-algebra generated by $A(C)$.

Since $C\subset A(C)$ and $A(C)\subset\sigma [A(C)]$ then $C\subset \sigma [A(C)]$. $\sigma(C)$ is the smallest $\sigma$-algebra containing $C$ thus $\sigma(C)\subset\sigma [A(C)]$.

For the other way, note that $C\subset \sigma(C)$. Since $\sigma(C)$ is also an algebra thus $A(C)\subset \sigma(C)$. $\sigma [A(C)]$ is the smallest algebra containing $[A(C)]$ thus $\sigma [A(C)]\subset \sigma(C)$.

Is my proof correct? In the second part I assumed that $A(C)$ is the smallest algebra containing $C$ that is why I got $A(C)\subset \sigma(C)$. I can't find the definition of "the algebra generated by $C$" anywhere. The definition I have is that the $\sigma$-algebra generated by a set $C$ is the smallest $\sigma$-algebra that contains $C$. So I assumed that "the algebra generated by $C$" is the smallest algebra that contains $C$. Am I right in assuming this? I hope somebody could clarify this for me.

Also, what is the smallest algebra containing a set $C$? Is it $\{\emptyset,C\}$? So is $\{\emptyset,C\}$ the algebra generated by $C$?

Thank you!

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That's right. Whenever you have "the $A$ generated by $B$", it almost always means "the smallest $A$ containing $B$" in some sense (usually there's some large object which his also $A$ and contains $B$ so that the statement even makes sense, or some adjective describing the way the object is constructed, that is, we know what operations on $B$ look like; in your case, the large object is the algebra $\mathcal P(X)$ of all subsets of $X$, but on the other hand, we don't need an universe to know what a ring of sets generated by a given family looks like, neither do we need that to say what a group freely generated by a set looks like).For your second point, it would depend on the universe of the algebra. If $C=X$ is the universe, then the algebra is $\{\emptyset,C\}$. In general, it is $\{\emptyset,C,X\setminus C, X\}$ where $X$ is the universe. ...