  In this paper we construct an invariant weighted Wiener measure associated to
the periodic derivative nonlinear Schr\"odinger equation in one dimension and
establish global well-posedness for data living in its support. In particular
almost surely for data in a Fourier-Lebesgue space ${\mathcal F}L^{s,r}(\T)$
with $s \ge \frac{1}{2}$, $2 < r < 4$, $(s-1)r <-1$ and scaling like
$H^{\frac{1}{2}-\epsilon}(\T),$ for small $\epsilon >0$. We also show the
invariance of this measure.
