Approximate weak amenability of certain Banach algebras

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.