  Given $n$ equidistant realisations of a L\'evy process $(L_t,\,t\ge 0)$, a
natural estimator $\hat N_n$ for the distribution function $N$ of the L\'evy
measure is constructed. Under a polynomial decay restriction on the
characteristic function $\phi$, a Donsker-type theorem is proved, that is, a
functional central limit theorem for the process $\sqrt n (\hat N_n -N)$ in the
space of bounded functions away from zero. The limit distribution is a
generalised Brownian bridge process with bounded and continuous sample paths
whose covariance structure depends on the Fourier-integral operator ${\cal
F}^{-1}[1/\phi(-\cdot)]$. The class of L\'evy processes covered includes
several relevant examples such as compound Poisson, Gamma and self-decomposable
processes. Main ideas in the proof include establishing pseudo-locality of the
Fourier-integral operator and recent techniques from smoothed empirical
processes.
