Corollary 11.47.8EA98BEE991F7

Proof. On an annulus, the radial flux formula (11.46.4), followed by convex smoothing and bounded Bedford–Taylor continuity, gives the displayed measure. Continuity of $C_ F'$ excludes boundary atoms. The non-pluripolar truncations discard $o$, and smoothness at infinity excludes hyperplane mass. The asymptotic slopes $0,1$ therefore give mass one. An affine profile has zero radial eigenvalue and hence zero top-degree measure. Finally, if $\delta =\| F-G\| _\infty $, subtracting $\delta $ from a competitor for $F$ gives a competitor for $G$, and conversely. The resulting value bound for the envelopes gives (11.47.9) off $o$, and continuity of the residuals extends it across both ends. $\square$