We study in this paper the space $L^\infty_0({\cal
S},M_a({\cal S}))$ of a locally compact semigroup ${\cal S}$. That
space consists of all $\mu$-measurable ($\mu\in M_a({\cal S})$)
functions vanishing at infinity, where $M_a({\cal S})$ denotes the
algebra of all measures with continuous translations. We introduce
an Arens multiplication on the dual $L^\infty_0({\cal S},M_a({\cal
S}))^*$ of $L^\infty_0({\cal S},M_a({\cal S}))$ under which
$M_a({\cal S})$ is an ideal. We then give some characterizations
for Arens regularity of $M_a({\cal S})$ and $L^\infty_0({\cal
S},M_a({\cal S}))^*$. As the main result, we show that $M_a({\cal
S})$ or $L^\infty_0({\cal S},M_a({\cal S}))^*$ is Arens regular if
and only if ${\cal S}$ is finite.