Lemma 5.43.16.151A1D744DE7 For every finite signed measure $\mu $ and Borel set $B\subset C$,
\[ \begin{gathered} r_*r^*\mu =d\mu ,\qquad \| r^*\mu \| _{\mathrm{TV}}=d\| \mu \| _{\mathrm{TV}},\\ r^*(\mathbf1_ B\mu )=\mathbf1_{r^{-1}(B)}r^*\mu , \qquad r^*\delta _ s=\sum _{r(x)=s}e_ x\delta _ x. \end{gathered} \]
For $v\in \mathrm{PSH}(C,\beta )$, writing $T_ v=\beta +\mathrm{dd}^{\mathrm{c}}v$ and $\mu _ v=\mathbf1_{\{ v=-\infty \} }T_ v$, one has
\[ T_{v\circ r}=r^*T_ v,\qquad \mu _{v\circ r}=r^*\mu _ v. \]
Images and inverse images of polar sets under $r$ are polar. Pullback and pushforward of positive measures commute with taking the atomic part.
Proof.
The trace of $g\circ r$ is $dg$, giving the pushforward identity. The same weighted fiber formula for Borel test functions follows from the monotone-class theorem and gives the restriction and atom formulas. The positive and negative Jordan parts of a signed measure have disjoint Borel carriers; their pullbacks are positive with disjoint inverse-image carriers. Each has $d$ times its original mass, which proves the total-variation identity.
Choose a local smooth $a$ with $\mathrm{dd}^{\mathrm{c}}a=\beta $; then $v+a$ is a subharmonic potential of $T_ v$. Its Riesz representation is a logarithmic potential plus a harmonic function. In coordinates $t=z^ e$, the identity
\[ \mathrm{dd}^{\mathrm{c}}_ z\log |z^ e-t|^2 =\sum _{z^ e=t}\operatorname {ord}_ z(z^ e-t)\delta _ z \]
shows that pulling back this potential gives the weighted measure above. Fubini's theorem applies locally: factorization of $z^ e-t$ into its $e$ linear factors gives a uniform local $L^1$ bound for these logarithmic kernels as $t$ varies on a smaller disk. The harmonic term has zero Hessian after pullback. This proves $T_{v\circ r}=r^*T_ v$. The pole set of $v\circ r$ is exactly $r^{-1}\{ v=-\infty \} $, so the restriction formula proves the polar identity.
Inverse images of polar sets are polar by composition of local subharmonic witnesses. For images, take local witnesses on the finitely many inverse-image disks of a small base disk. Off the branch values, the sum of their values over the inverse branches is subharmonic and is $-\infty $ wherever any witness has a pole. It is locally bounded above and is not identically $-\infty $. Its extension across the branch values is subharmonic. Adding the finitely many branch values to the resulting polar set still gives a polar set. This proves the image assertion locally, hence globally.
The pullback of a nonatomic measure has no atoms by the atom formula. The pushforward of a nonatomic measure has no atoms because every fiber is finite. Atomic measures pull back and push forward to atomic measures. Splitting a positive measure into these two parts proves the last assertion.
$\square$
Pome
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