Let $(X,\mathscr F,\mu)$ be a measurable space with a positive measure $\mu$. Let us call the set $F\in\mathscr F$ full if $\mu(F^c) = 0$. I wonder if there exists the smallest full set $F\in \mathscr F$ in the following sense:
if $B\supseteq F$, $B\in \mathscr F$ then $\mu(B^c) = 0$;
if $F'\in \mathscr F$ is another set satisfying 1. then $\mu(F\setminus F') = 0$, i.e. $F$ lies in $F'$ almost completely.
If the existence does not hold for the general setting, would it be sufficient to assume that $\mu$ is $\sigma$-finite and/or that $\mathscr F$ is countably generated?