Definition 7.2.1254CDCCCAB5E

Definition 7.2.1.254CDCCCAB5E Suppose that $r\ge 1$, and let $u$ be a locally bounded plurisubharmonic function on $\Omega $. Let $uT$ be the current of order zero obtained by multiplying the coefficient measures of $T$ by the locally bounded Borel function $u$ (section 5.3), and define

For an integer $1\le q\le r$ and locally bounded plurisubharmonic functions $u_1,\ldots ,u_ q$ on $\Omega $, define recursively

Here $\mathrm{dd}^{\mathrm{c}}u_2\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ q\wedge T$ is a closed positive current, hence of order zero, by proposition 7.2.2 below applied to $q-1$ factors; so its product with $u_1$ is defined as before. The product with no factors is $T$. When $T=1$ (so that $r=n$), $q=n$ and $u_1=\cdots =u_ n=u$, the resulting positive current of bidegree $(n,n)$ is a positive measure, the Monge–Ampère measure $(\mathrm{dd}^{\mathrm{c}}u)^ n$ of $u$.