  In this paper, we prove that for each closed differential form $u \in
L^1(\mathbb{R}^N,(\mathbb{R}^N)^{\ast} \wedge ... \wedge
(\mathbb{R}^N)^{\ast})$, which is almost in $L^{\infty}$ in the sense that \[
  \int_{\{y \in \mathbb{R}^N \colon \vert u(y) \vert \geq L \}} \vert u(y)
\vert dy < \varepsilon \] for some $L>0$ and a small $\varepsilon >0$, we may
find a closed differential form $v$, such that $\Vert u - v \Vert_{L^1}$ is
again small, and $v$ is, in addition, in $L^{\infty}$ with a bound on its
$L^{\infty}$ norm depending only on $N$ and $L$. In particular, the set $\{ v
\neq u\}$ has measure at most $C \varepsilon$. We then look at applications of
this theorem. We are able to prove that the $\mathcal{A}$-$p$-quasiconvex hull
of a set does not depend on $p$. Furthermore, we can prove a classification
theorem for $\mathcal{A}$-$\infty$-Young measures.
