  It is shown that for a locally compact group $G$, if $L^{1}(G)^{**}$ is
approximately weakly amenable, then $M(G)$ is approximately weakly amenable.
Then, new notions of approximate weak amenability and approximate cyclic
amenability for Banach algebras are introduced. Bounded
$\omega^{*}$-approximately weakly [cyclic] amenable $\ell^{1}$-Munn algebras
are characterized.
