  In functional analysis, there are different notions of limit for a bounded
sequence of $L^1$ functions. Besides the pointwise limit, that does not always
exist, the behaviour of a bounded sequence of $L^1$ functions can be described
in terms of its weak-$\star$ limit or by introducing a measure-valued notion of
limit in the sense of Young measures. Working in Robinson's framework of
analysis with infinitesimals, we show that for every bounded sequence
$\{z_n\}_{n \in \mathbb{N}}$ of $L^1$ functions there exists a function of a
hyperfinite domain (i.e.\ a grid function) that represents both the
weak-$\star$ and the Young measure limits of the sequence. This result has
relevant applications to the study of nonlinear PDEs. We discuss the example of
an ill-posed forward-backward parabolic equation.
