  We prove absolute continuity of Gaussian measures associated to complex
Brownian bridges under certain gauge transformations. As an application we
prove that the invariant measure for the periodic derivative nonlinear
Schr\"odinger equation obtained by Nahmod, Oh, Rey-Bellet and Staffilani in
[20], and with respect to which they proved almost surely global
well-posedness, coincides with the weighted Wiener measure constructed by
Thomann and Tzvetkov [24]. Thus, in particular we prove the invariance of the
measure constructed in [24].
