We study the one-dimensional periodic derivative nonlinear Schrödinger equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion ({textstyle int }h_k), (kin {mathbb {Z}}_{+}). In each ({textstyle int }h_{2k}) the term with the highest regularity involves the Sobolev norm (dot{H}^{k}({mathbb {T}})) of the solution of the DNLS equation. We show that a functional measure on (L^2({mathbb {T}})), absolutely continuous w.r.t. the Gaussian measure with covariance (({mathbb {I}}+(-varDelta )^{k})^{-1}), is associated to each integral of motion ({textstyle int }h_{2k}), (kge 1).