# Question 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!

EDHF2A The Asker · Other Mathematics

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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