  Inclusive J/$\psi$ production is studied in minimum-bias proton-proton
collisions at a centre-of-mass energy of $\sqrt{s}$ = 5.02 TeV by ALICE at the
CERN LHC. The measurement is performed at mid-rapidity ($|y| < 0.9$) in the
dielectron decay channel down to zero transverse momentum $p_{\rm T}$, using a
data sample corresponding to an integrated luminosity of $L_{\rm int} = 19.4
\pm$ 0.4 nb$^{-1}$. The measured $p_{\rm T}$-integrated inclusive J/$\psi$
production cross section is d$\sigma$/d$y$ = 5.64 $\pm$ 0.22 (stat.) $\pm 0.33$
(syst.) $\pm 0.12$ (lumi.) $\mu$b. The $p_{\rm T}$-differential cross section
d$^{2} \sigma$/d$p_{\rm T}$d$y$ is measured in the $p_{\rm T}$ range 0$-$10
GeV/$c$ and compared with state-of-the-art QCD calculations. The J/$\psi$
$\langle p_{\rm T} \rangle$ and $\langle p_{\rm T}^{2} \rangle$ are extracted
and compared with results obtained at other collision energies.
