Proposition 5.46.55AAB9E80546A

Proof. By proposition 5.43.18, $u\circ \pi \in \mathrm{PSH}(Y,\pi ^*\theta )$. Change of variables on the locus where $\pi $ is an isomorphism, whose complement has Lebesgue measure zero in both manifolds, gives $\pi _*(u\circ \pi )=u$ as distributions and $\pi _*\pi ^*\theta =\theta $ as currents. Since pushforward by a proper map commutes with $\mathrm{dd}^{\mathrm{c}}$, we get $\pi _*T_ Y=T$.

For any Borel set $B\subset X$, the coefficient measures of $\pi _*T_ Y$ are pushforwards of the coefficient measures of $T_ Y$, multiplied by the smooth coefficients that arise from pulling back a test form. Restriction therefore gives $\pi _*(\mathbf1_{\pi ^{-1}(B)}T_ Y)=\mathbf1_ B\pi _*T_ Y$. Applying this to the Borel set $S$ proves (5.46.6); closedness of the restricted currents is not used. $\square$