  Via the hierarchy of correlations, we study the strongly interacting
Fermi-Hubbard model in the Mott insulator state and couple it to a Markovian
environment which constantly monitors the particle numbers $\hat
n_\mu^\uparrow$ and $\hat n_\mu^\downarrow$ for each lattice site $\mu$. As
expected, the environment induces an imaginary part $\gamma$ (i.e., decay rate)
of the quasi-particle frequencies
$\omega_{\mathbf{k}}\to\omega_{\mathbf{k}}-i\gamma$ and tends to diminish the
correlations between lattice sites. Surprisingly, the environment does also
steer the state of the system on intermediate time scales
$\mathcal{O}(1/\gamma)$ to a pre-thermalized state very similar to a quantum
quench (i.e., suddenly switching on the hopping rate $J$). Full thermalization
occurs via local on-site heating and takes much longer.
