  In this paper, we study the weak amenability of weighted measure algebras and
prove that $M(G, \omega)$ is weakly amenable if and only if $G$ is discrete and
every bounded quasi-additive function is inner. We also study the weak
amenability of $L^1(G, \omega)^{**}$ and $M(G, \omega)^{**}$ and show that the
weak amenability of theses Banach algebras are equivalent to finiteness of $G$.
This gives an answer to the question concerning weak amenability of $L^1(G,
\omega)^{**}$ and $M(G, \omega)^{**}$.
