  We give equivalent conditions for the existence of generalized moments of a
L\'evy process $(X_t)_{t\geq 0}$. We show, in particular, that the existence of
a generalized $g$-moment is equivalent to uniform integrability of
$(g(X_t))_{t\in [0,1]}$. As an application, it turns out that certain functions
of a L\'evy process which are integrable and local martingales are already true
martingales.
