  We establish lower semicontinuity results for perimeter functionals with
measure data on $\mathbb{R}^n$ and deduce the existence of minimizers to these
functionals with Dirichlet boundary conditions, obstacles, or
volume-constraints. In other words, we lay foundations of a perimeter-based
variational approach to mean curvature measures on $\mathbb{R}^n$ capable of
proving existence in various prescribed-mean-curvature problems with measure
data. As crucial and essentially optimal assumption on the measure data we
identify a new condition, called small-volume isoperimetric condition, which
sharply captures cancellation effects and comes with surprisingly many
properties and reformulations in itself. In particular, we show that the
small-volume isoperimetric condition is satisfied for a wide class of
$(n{-}1)$-dimensional measures, which are thus admissible in our theory. Our
analysis includes infinite measures and semicontinuity results on very general
domains.
