  We construct invariant measures associated to the integrals of motion of the
periodic derivative nonlinear Schr\"odinger equation (DNLS) for small data in
$L^2$ and we show these measures to be absolutely continuous with respect to
the Gaussian measure. The key ingredient of the proof is the analysis of the
gauge group of transformations associated to DNLS. As an intermediate step for
our main result, we prove quasi-invariance with respect to the gauge maps of
Gaussian measures on $L^2$.
