  One way to interpret smoothness of a measure in infinite dimensions is
quasi-invariance of the measure under a class of transformations. Usually such
settings lack a reference measure such as the Lebesgue or Haar measure, and
therefore we can not use smoothness of a density with respect to such a
measure. We describe how a functional inequality can be used to prove
quasi-invariance results in several settings. In particular, this gives a
different proof of the classical Cameron-Martin (Girsanov) theorem for an
abstract Wiener space. In addition, we revisit several more geometric examples,
even though the main abstract result concerns quasi-invariance of a measure
under a group action on a measure space.
