Q:

Is is always true that $E[Z|\mathscr{G}]1_{\{Z=0\}}=0$ a.s. ？  And why? 

Is is always true that $E[Z|\mathscr{G}]1_{\{Z=0\}}=0$ a.s. ？ 
And why? 

A:

Not true.  If $Z$ is independent of $\mathcal G$ then LHS is $EZ$ on the set $\{Z=0\}$ (which may have positive measure).