  A natural question in mathematical general relativity is how the ADM mass
behaves as a functional on the space of asymptotically flat 3-manifolds of
nonnegative scalar curvature. In previous results, lower semicontinuity has
been established by the first-named author for pointed $C^2$ convergence, and
more generally by both authors for pointed $C^0$ convergence (all in the
Cheeger--Gromov sense). In this paper, we show this behavior persists for the
much weaker notion of pointed Sormani--Wenger intrinsic flat ($\mathcal{F}$)
volume convergence, under natural hypotheses. We consider smooth manifolds
converging to asymptotically flat local integral current spaces (a new
definition), using Huisken's isoperimetric mass as a replacement for the ADM
mass. Along the way we prove results of independent interest about convergence
of subregions of $\mathcal{F}$-converging sequences of integral current spaces.
