This article discusses the perturbation of a non-symmetric Dirichlet form,
$({\cal E},{\cal D(E)})$, by a signed smooth measure $\mu$, where $\mu$=$\mu_{1}-\mu_{2}$ with $\mu_{1}$ and $\mu_{2}$ being smooth
measures. It gives a sufficient condition for the perturbed form
$({{\cal E}}^{\mu},{{\cal D}}({{\cal E}}^{\mu}))$ (for some $\alpha_0\geq 0)$ to be a coercive closed form.