  Given a function $f\in C(\mathbb{R}^d)$ of linear growth, we give a new way
of representing accumulation points of \begin{equation} \int_\Omega
f(v_i(z))d\mu(z), \end{equation} where $\mu\in \mathcal{M}^+(\Omega)$, and
$(v_i)_{i\in \mathbb{N}}\subset L^1(\Omega,\mu)$ is norm bounded. We call such
representations "generalised Young Measures". With the help of the new
representations, we then characterise these limits when they are generated by
gradients, i.e. when $v_i = Du_i$ for $u_i\in W^{1,1}(\Omega,\mathbb{R}^m)$,
via a set of integral inequalities.
