Inclusion of $\sigma$-algebras Given $X_1, X_2, X_3$ three random variables taking values on $\mathbb{R}$, I want to see whether $\mathcal{F}_{X_1X_2}\subset \mathcal{F}_{X_1X_2X_3}$ or $\mathcal{F}_{X_1X_2}\supset \mathcal{F}_{X_1X_2X_3}$. My intuition tells me that $\mathcal{F}_{X_1X_2X_3}$ is greater because it contains more information, but I don't know how to justify it... Help, please? Thanks
 A: In general, given real-valued random variables $X$ and $Y$ there is no relation between the sigma algebras for $F_{X}, F_{Y}$ and $F_{XY}$. Applying this to $X=X_1X_2$ and $Y=X_3$, we conclude that it is possible to have $F_{X_1X_2}\not\subseteq F_{X_1X_2X_3}$ and $F_{X_1X_2}\not\supseteq F_{X_1X_2X_3}$.
It helps to look at an example. Let $\Omega=\{(1,1),(1,2),(2,1),(2,2)\}$ with the powerset $\sigma$-algebra and uniform measure, and let $X(\omega_1,\omega_2)=\omega_1$ and $Y(\omega_1,\omega_2)=\omega_2$. You can check that none of $F_X,F_Y$ and $F_{XY}$ include each other. Since $\sigma$-algebras on a finite set, $\Omega$,  are equivalent to partitions of $\Omega$, I find this is best illustrated as shown
$$
\mathcal F_X:\begin{array}{c|c}1 & 1 \\\hline \color{red}2 & \color{red}2\end{array}
\qquad
\mathcal F_Y:\begin{array}{c|c}1 & \color{red}2 \\\hline 1 & \color{red}2\end{array}
\qquad 
\mathcal F_{XY}:\begin{array}{c|c}1 & \color{red}2 \\\hline \color{red}2 & \color{orange}4\end{array}
$$
