  This paper deals with the positive mass theorem and the existence of
isoperimetric sets on $3$-manifolds endowed with continuous complete metrics
having nonnegative scalar curvature in a suitable weak sense.
  We prove that if the manifold has an end that is $C^0$-locally asymptotically
flat, and the metric is the local uniform limit of smooth metrics with
vanishing lower bounds on the scalar curvature outside a compact set, then
Huisken's isoperimetric mass is nonnegative. This addresses a version of a
recent conjecture of Huisken about positive isoperimetric mass theorems for
continuous metrics satisfying $R_g\geq 0$ in a weak sense. As a consequence,
any fill-in of a truncation of a Schwarzschild space with negative ADM mass has
nonnegative isoperimetric mass.
  Moreover, in case the whole manifold is $C^0$-locally asymptotically flat and
the metric is the local uniform limit of smooth metrics with vanishing lower
bounds on the scalar curvature outside a compact set, we prove that, as a large
scale effect, isoperimetric sets with arbitrarily large volume exist.
