We construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger 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 $ℱL^{s,r}\left(T\right)$ with $s\ge \frac{1}{2},2<r<4,(s-1)r<-1$ and scaling like $H^{\frac{1}{2}-ϵ}\left(𝕋\right)$, for small $ϵ>0$. We also show the invariance of this measure.