If $\chi$ is a family of random processes, then $\chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$, the localized class of $\chi$, will denote the family of processes wich are locally in $\chi$ for $(\Bbb{F},\Bbb{P})$ this means that $X\in \chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$ if there exists a nondecreasing sequence of stopping times $(\tau_n)_n$ s.t. $\lim_{n\rightarrow \infty} \tau_n=\infty$ and $\Bbb{1}_{\tau_n>0} X^{\tau_n}\in \chi$ for all $n$. we call this $(\tau_n)_n$ a localizing sequence
Now I would like to show that $$\left(\chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})\right)_{\operatorname{loc}}(\Bbb{F},\Bbb{P})=\chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$$
My idea was the following:
$\subseteq~~$If I take $X\in\left(\chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})\right)_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$ then there exists a localising sequence $(\tau_n)_n$ s.t. $\Bbb{1}_{\tau_n>0} X^{\tau_n}\in \chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$. But this means that there exists another localizing sequence $(\rho_m)_m$ s.t. $\Bbb{1}_{\rho_m>0}\left(\Bbb{1}_{\tau_n>0} X^{\tau_n}\right)^{\rho_m}\in \chi$. Now define $\sigma_{m,n}=\rho_m\wedge\tau_n$ then this is a sequence of stopping times s.t. $\lim_{m,n\rightarrow \infty}\sigma_{m,n}=\infty$. Furhtermore $\Bbb{1}_{\sigma_{m,n}>0}X^{\sigma_{m,n}}=\Bbb{1}_{\rho_m>0}\left(\Bbb{1}_{\tau_n>0} X^{\tau_n}\right)^{\rho_m}\in \chi$. Hence $X\in \chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$.
$\supseteq~~$ Let $X\in \chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$. Then there exists a localising sequence $\tau_n$ s.t. $\Bbb{1}_{\tau_n>0} X^{\tau_n}\in \chi\subset \chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$. So $X\in\left(\chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})\right)_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$.
Does this work?