  In this paper, as the first step towards classification of simple weight
modules with finite dimensional weight spaces over Witt algebras $W_n$, we
explicitly describe supports of such modules. We also obtain some descriptions
on the support of an arbitrary simple weight module over a $\Z^n$-graded Lie
algebra $\mathfrak{g}$ having a root space decomposition
$\oplus_{\alpha\in\Z^n}\mathfrak {g}_\alpha$ with respect to the abelian
subalgebra $\mathfrak {g}_0$, with the property $[\mathfrak{g}_\alpha,\mathfrak
{g}_\beta]= \mathfrak {g}_{\alpha+\beta}$ for all $\alpha,\beta\in\Z^n$,
$\alpha\neq \beta$ (this class contains the algebra $W_n$).
