$\sigma\mathcal C$ is the $\sigma$-algebra generated by $\mathcal C$. Show $\sigma\mathcal C\subset\sigma\mathcal D$ if $\mathcal C\subset\mathcal D$. If $\mathcal{C}$ and $\mathcal{D}$ are two collections of subsets of $E$. How do I prove the following:
$$\mathcal{C}\subset\mathcal{D}\implies\sigma\mathcal{C}\subset\sigma\mathcal{D}?$$

You could also use the fact that $\sigma\mathcal{A}$ is the smallest $\sigma$-algebra containing $\mathcal{A}$. If you know this fact then you can derive the desired result in a few easy steps:


*

*$\sigma\mathcal{D}$ is the smallest $\sigma$-algebra containing $\mathcal{D}$; 

*$\mathcal{C}$ is contained in $\mathcal{D}$;

*$\sigma\mathcal{D}$ is a $\sigma$-algebra containing $\mathcal{C}$;

*$\sigma\mathcal{C}$ is the . . .

*Therefore $\sigma\mathcal{C}$ is contained in $\sigma\mathcal{D}$.

