if the variable depends on $\theta$, like $x>\theta$, indicator s.t $I_{(\theta,\infty)}{x}$ be used, what if $x>0$, should I still use indicator?

Question

For finding the minimal sufficient statistic, if the variable depends on the unknown parameter, like $x>\theta$, we should use indicator s.t $I_{(\theta,\infty)}{x}$, but if the variable is just like $x>0$, should I still use indicator? Thanks for your help.