A "foliation current" in the sense of Ruelle-Sullivan (https://www.math.stonybrook.edu/~ebedford/PapersForM655/RS.pdf) is essentially a closed subset of a manifold foliated by equidimensional oriented submanifolds along with a transverse measure, so integration makes sense: just integrate along each leaf, then integrate via the transverse measure.
I was wondering (although I don't have much hope) if a weak-* limit of a sequence of foliation currents is again a foliation current? Perhaps we would need to put some dimensional assumptions to exclude bad formations of singularities, or expand our notion of foliation current to a more singular one.
