Continuity of quaternionic Monge–Ampère measures

Statement

Let $u_m,u$ be continuous quaternionic plurisubharmonic functions on $\Omega\subseteq\mathbb H^n$ . If $u_m\to u$ uniformly on compact subsets, then

\operatorname{MA}_{\mathbb H}(u_m) \rightharpoonup \operatorname{MA}_{\mathbb H}(u)

weakly as Borel measures. The same statement holds for mixed measures when each potential converges locally uniformly.

Significance

This theorem both characterizes the nonsmooth quaternionic Monge–Ampère measure and makes it stable under approximation. It is the quaternionic counterpart of Aleksandrov's continuity theorem for convex Hessian measures and of the Chern–Levine–Nirenberg continuity theory in complex analysis.

Application to convex bodies

Convergence of convex bodies in the Hausdorff metric is equivalent to locally uniform convergence of their support functions. The theorem therefore supplies the continuity part of the pluripotential construction of valuations.

References

Semyon Alesker, “Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables,” Bulletin des Sciences Mathématiques 127 (2003), 1–35. arXiv record. Relevant: Theorem 3.4.
Semyon Alesker, “Valuations on convex sets, non-commutative determinants, and pluripotential theory,” Advances in Mathematics 195 (2005), 561–595. arXiv record.