Bedford–Taylor theory

Remarks

The main facts of the theory are these. The product $dd^c u \wedge T$ is again a closed positive current, which makes the induction possible. When the $u_j$ are of class $C^2$ the product agrees with the wedge product of smooth forms, so $(dd^c u)^n$ agrees with the complex Monge–Ampère operator. If $u_j^{(m)}$ decreases to $u_j$ as $m \to \infty$, for each $j$, then the products of the $u_j^{(m)}$ converge weakly to the product of the $u_j$. The product is symmetric in $u_1, \dots, u_k$. Example: for $n = 1$ and $T = 1$, $dd^c u = \Delta u$ is the Riesz measure of the subharmonic function $u$, which is defined for every subharmonic $u$. Non-example: the definition does not apply to $u = \log\abs{z_1}$ on $\C^2$, which is not locally bounded. For unbounded plurisubharmonic functions in general, $(dd^c u)^n$ cannot be defined as a measure with good continuity properties.