I have a question. If I have a sigma algebra $\sigma(X_1)$ generated by $X_1$ where $X_1 \subset 2^P $ . And say $X_2$ is another sigma algebra where $X_2 \subset 2^P $.
I was reading that if $X_1$ is contained in $\sigma(X_2)$, then automatically $\sigma(X_1)$ is contained in $\sigma(X_2)$. But I am not sure why.
Could someone explains?
Another way is see:
$\sigma(X_{1})=\displaystyle\bigcap\{P: P~\text{is a}~\sigma-\text{algebra that containing}~X_{1}\}$. Now $\sigma(X_{2})\in\bigcap\{P: P~\text{is a}~\sigma-\text{algebra that containing}~X_{1}\}$, so $\sigma(X_{1})\subseteq\sigma(X_{2})$.