Notation: If $X$ and $Y$ are sets, $\sigma(X)$ and, $\sigma(Y)$ are the $\sigma$-algebras generated by those sets respectively and $L^0(\sigma(X))$ and,$L^0(\sigma(Y))$ are the sets of measurable functions with respect to the $\sigma$-algebras generated by $X$ and $Y$ respectively.
Suppose I know that $L^0(\sigma(X))\subset L^0(\sigma(Y))$ then can I conclude that $X\subseteq Y$?
Consider $A\in \sigma (X)$, then the characteristic function of $A$ is $\sigma(X)$-measurable and based on the assumption, is $\sigma(Y)$-measurable, which implies $A\in \sigma(Y)$. So $\sigma(X)\subseteq \sigma(Y)$. However nothing can be said about $X$ and $Y$. For instance consider $X=\{\emptyset, A,\Omega\}$ and $Y=\{\emptyset, A^c,\Omega\}$. They both generate $\sigma(X)=\sigma(Y)$ but neither $X\subseteq Y$ not $Y\subseteq X$.