Lemma 5.27.263242D4BA1F13

Proof. Write $t=\log |z|^2$ and approximate $\max \{ t,\log r^2\} $ uniformly by smooth convex increasing functions $F_\epsilon (t)$ whose derivatives equal zero below $\log r^2-\epsilon $ and one above $\log r^2+\epsilon $. Their compositions with $t$ extend smoothly across zero. Since $\mathrm{dd}^{\mathrm{c}}t$ has rank $n-1$ off zero, the chain rule gives

The integral of $\mathrm d^{\mathrm c}t\wedge (\mathrm{dd}^{\mathrm{c}}t)^{n-1}$ over each sphere is one. Indeed, $\mathrm d^{\mathrm c}t$ integrates to one on the Hopf circle, and the induced form on $\mathbb P^{n-1}$ is $\mathrm{dd}^{\mathrm{c}}\log (1+|w|^2)$; its top power has density $(n-1)!\pi ^{-(n-1)}(1+|w|^2)^{-n}$ on the affine chart, whose integral is one. Thus each displayed radial measure has total mass $\int ( (F_\epsilon ')^ n)'\, \mathrm dt=1$. Its support shrinks to the sphere, and its rotation invariance identifies the limit as $\sigma _ r$. Uniform bounded-potential convergence identifies that limit with the product in the statement. Non-pluripolarity follows from theorem 5.27.1. $\square$