  We give new proofs of certain equivalent conditions for the existence of
generalized moments of a L\'evy process $(X_t)_{t\geq 0}$; in particular, the
existence of a generalized $g$-moment is equivalent to the uniform
integrability of $(g(X_t))_{t\in [0,1]}$. As a consequence, certain functions
of a L\'evy process which are integrable and local martingales are already true
martingales. Our methods extend to moments of stochastically continuous
additive processes, and we give new, short proofs for the characterization of
lattice distributions and the transience of L\'evy processes.
