  In [17], the authors introduced the space $GBV_\star(A)$ to minimise a class
of functionals whose study is motivated by fracture mechanics. In this paper,
we extend the definition of $GBV_\star(A)$ to the vectorial case, introducing
the space $GBV_\star(A;\mathbb{R}^k)$. We study the main properties of
$GBV_\star(A;\mathbb{R}^k)$ and prove a lower semicontinuity result useful for
minimisation purposes. With the Direct Method in mind, we adapt the arguments
of [17] to show that minimising sequences in $GBV_\star(A;\mathbb{R}^k)$ can be
modified to obtain a minimising sequence converging $\mathcal{L}^d$-a.e in $A$.
