  We study the Monge-Amp\` ere operator within the framework of Dinh-Sibony's
intersection theory defined via density currents. We show that if $u$ is a
plurisubharmonic function belonging to the Blocki-Cegrell class, then the
Dinh-Sibony $n$-fold self-product of $\text{dd}^c u$ exists and coincides with
the classically defined Monge-Amp\`ere measure $(\text{dd}^c u)^n$.
