why is pushforward of a kähler current kähler?

Question

A Kähler current $T$ on a compact complex manifold is a positive closed $(1,1)$-current such that $T > \psi$ for a positive (not necessarily closed) $(1,1)$-form $\psi$. Let $f: X \to Y$ be a proper map, $dim X = dim Y$, then $f_* T$ is a closed positive $(1,1)$-current. How does one see that $f_* T$ is bounded below by a positive form and so is a Kähler current?