Measurability of a random varaiable over sigma-field generated by a countable family of random variables Let $(X_\alpha)_{\alpha\in A}$ be a family of random variables and $\mathcal{F}=\sigma(X_\alpha:\alpha\in A)$. If $Z$ is $\mathcal{F}$ measurable r.v, show that there is a countable subset $\Gamma\subset A$ such that $\sigma(Z)\subset\sigma(X_\alpha :\alpha\in\Gamma)$.  
My initial thoughts was to consider $\{Z<q\}$ for any $q\in \mathbb{Q}$ and show that there is $\alpha$ such that $\{Z<q\}\in\sigma(X_\alpha)$, and mabye invoke the monotone class theorem on $\mathcal{G}:=\{\{Z<q\}| q\in\mathbb{Q} \space \& \space \{Z<q\}\in\sigma(X_\alpha) \text{ for some}\alpha\in A\}$ but had no luck there as I can't see how to show $\{Z<q\}\in\sigma(X_\alpha)$.  
Another idea was to try a Zorn's lemma argument to find $\Gamma$, so applying it to $\mathcal{P}:=\{\Gamma|\Gamma\subset A\space \&\space |\Gamma|\leq\aleph_0\space \&\space \sigma(Z)\cap\sigma(X_\alpha:\alpha\in\Gamma)\neq\emptyset\}$ but I couldn't get anywhere there as well. 
I am really stuck on this and would appreciate any help. Thanks in advance.
 A: Define $\mathcal A:=\bigcup_{\alpha\in A}\sigma\left(X_\alpha\right)$ and let 
$$\mathcal B:=\left\{A \subset \Omega \mid \mbox{there exists a countable subset }\Gamma\subset A\mbox{ such that }A\in\sigma\left(X_\alpha,\alpha\in\Gamma\right)\right\}.$$ 
Then the collection $\mathcal B$ is a $\sigma$-algebra containing $\mathcal A$. This proves that $\mathcal F$ is contained in $\mathcal  B$. Now, if $r$ is a rational number and $Z$ is $\mathcal F$-measurable, then $\left\{Z\lt r\right\}$ belongs to $\mathcal F$ hence to $\mathcal B$: there exists a countable set $\Gamma_r$ such that $\left\{Z\lt r\right\}\in \sigma\left(X_\alpha,\alpha \in \Gamma_r\right)$. defining $\Gamma:=\bigcup_{r\in\mathbb Q} \Gamma_r$, we get that for any rational number $r$, 
$ \left\{Z\lt r\right\}\in \sigma\left(X_\alpha,\alpha \in \Gamma\right)
$ from which it follows that $\left\{Z\lt t\right\}\in \sigma\left(X_\alpha,\alpha \in \Gamma\right)
$ for any real number $t$ hence $\sigma(Z)\in \sigma\left(X_\alpha,\alpha \in \Gamma\right)$.
