  This study uses the ideas of \cite{Rieffel} to provide the dual of
$L^1(\mu,X)$ in the positive and $\sigma-$ finite cases. This results in
elegant necessary and sufficient criteria for weak compactness in
$L^1(S,\mu,X)$ in the $\sigma-$finite case, using the ideas of \cite{RuessL1}
and \cite{Cooper}. Finally, the result of \cite{NeervenLNM} is extended to
compute the sun-dual of $L^1(\re,X)$ with respect to the canonical translation
semigroup, dropping the approximation property from $X^*,$, which is applied to
obtain almost periodicity for integrals of non-smooth functions. Moreover, for
evolution semigroups, it is shown that weak compactness of the orbits implies
strong stability.
