Proof. [00SM]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Proof.

If $u$ is smooth, then

MA(u)(K)=\int_{K}\det(D^{2}u)dx_{1}\ldots dx_{n}=\frac{s^{n}}{\pi^{n}n!}\int_{(s^{-1}\text{Log})^{-1}(K)}(\sqrt{-1}\partial\bar{\partial}(u\circ s^{-1}\text{Log}))^{n}.

Since $u\in C^{0}$ , and both the real and complex MA operators are weakly continuous with respect to $C^{0}$ -limits, this equality passes to general $u$ . ∎