  Given an open set \Omega\subset\R^m and n>1, we introduce the new spaces
GB_nV(\Omega) of Generalized functions of bounded higher variation and
GSB_nV(\Omega) of Generalized special functions of bounded higher variation
that generalize, respectively, the space B_nV introduced by Jerrard and Soner
and the corresponding SB_nV space studied by De Lellis. In this class of
spaces, which allow the description of singularities of codimension n, the
distributional jacobian Ju need not have finite mass: roughly speaking,
finiteness of mass is not required for the (m-n)-dimensional part of Ju, but
only finiteness of size. In the space GSB_nV we are able to provide compactness
of sublevel sets and lower semicontinuity of Mumford-Shah type functionals, in
the same spirit of the codimension 1 theory.
