Algebra generated is a subset of sigma algebra generated Let $\mathcal C$ be a set of subsets of a sample space $\Omega$. I want to show that $a($$\mathcal C$$)$ $\subset$ $\sigma$$($$\mathcal C$$)$ and $\sigma$$(a$$($$\mathcal C$$)$$)$ $=$ $\sigma$$($$\mathcal C$$)$. I know definitions of each but can't get my head around these facts..
Anyone care to help?

$a(C)$ is the smallest algebra containing $C$, and $\sigma(C)$ is the smallest sigma-algebra containing $C$. Since all sigma-algebras are in particular algebras, we clearly see $a(C) \subset \sigma(C)$.
Now $\sigma(a(C))$ is the smallest sigma algebra containing $a(C)$. Since $a(C)$ contains $C$, $\sigma(a(C))$ is a sigma algebra containing $C$ and so $\sigma(C) \subset \sigma(a(C))$. Conversely, as we just showed, $\sigma(C)$ is a sigma algebra that contains $a(C)$ and so $\sigma(a(C)) \subset \sigma(C)$.
In summary, the trick is to just play around with the definitions, with emphasis on the use of minimality conditions.
