  Using a local perspective, we introduce mean dimension pairs and give
sufficient conditions of when every non-trivial factor of a continuous group
action of a sofic group $G$ has positive mean dimension. In addition, we prove
that the mean dimension map is Borel, and that the set of subshifts with
completely positive mean dimension of $[0,1]^G$, the full $G$-shift on the
interval, is a complete coanalytic set in the set of all subshifts (hence not
Borel). Our results are new even when the acting group is $\mathbb{Z}$.
