Let ( F t ) t 0 ( F t ) t 0 and ( G t ) t 0 ( G

Let ${\left(F_t\right)}_{t⩾0}$ and ${\left(G_t\right)}_{t⩾0}$ be any two filtrations and define $F_{\infty }=\sigma \left({\bigcup }_{t⩾0}{F}_t\right)$ and $G_{\infty }=\sigma \left({\bigcup }_{t⩾0}{G}_t\right)$. Show that 

Ft Gt (Vt > 0) if, and only if, Foo Goo