  We provide relaxation for not lower semicontinuous supremal functionals of
the type $W^{1,\infty}(\Omega;\mathbb R^d) \ni u \mapsto\supess_{ x \in
\Omega}f(\nabla u(x))$ in the vectorial case, where $\Omega\subset \mathbb R^N$
is a Lipschitz, bounded open set, and $f$ is level convex. The connection with
indicator functionals is also enlightened, thus extending previous lower
semicontinuity results in that framework. Finally we discuss the
$L^p$-approximation of supremal functionals, with non-negative, coercive
densities $f=f(x,\xi)$, which are only $\L^N \otimes \B_{d \times
N}$-measurable.
