  We discuss Lebesgue spaces $\mathcal{L}^p([a,b],E)$ of Lusin measurable
vector-valued functions and the corresponding vector spaces $AC_{L^p}([a,b],E)$
of absolutely continuous functions. These can be used to construct Lie groups
$AC_{L^p}([a,b],G)$ of absolutely continuous functions with values in an
infinite-dimensional Lie group $G$. We extend the notion of $L^p$-regularity of
infinite-dimensional Lie groups introduced by Gl\"ockner to this setting and
adopt several results and tools.
