Weak amenability of weighted measure algebras and their second duals

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 $L^1(G,\omega {)}^{\ast \ast }$ and $M(G,\omega {)}^{\ast \ast }$ 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 $L^1(G,\omega {)}^{\ast \ast }$ and $M(G,\omega {)}^{\ast \ast }$.