  We prove a fractional version of Poincar\'e inequalities in the context of
$\R^n$ endowed with a fairly general measure. Namely we prove a control of an
$L^2$ norm by a non local quantity, which plays the role of the gradient in the
standard Poincar\'e inequality. The assumption on the measure is the fact that
it satisfies the classical Poincar\'e inequality, so that our result is an
improvement of the latter inequality. Moreover we also quantify the tightness
at infinity provided by the control on the fractional derivative in terms of a
weight growing at infinity. The proof goes through the introduction of the
generator of the Ornstein-Uhlenbeck semigroup and some careful estimates of its
powers. To our knowledge this is the first proof of fractional Poincar\'e
inequality for measures more general than L\'evy measures.
