The aim of this paper is to develop a theory of decomposition
in the weighted modulation spaces $M_{p,q}^{s,W}$ with
$0 < p,q \le \infty$, $s \in \mathbb{R}$ and $W \in A_{\infty}$, where
$W$ belongs to the class of $A_{\infty}$ defined by Muckenhoupt.
For this purpose we shall define molecules for the modulation
spaces. As an application we give a simple proof of the boundedness
of the pseudo-differential operators with symbols in $M_{\infty,\min(1,p,q)}^{0}$.
We shall deal with dual spaces as well.