Capacity associated to a positive closed current

Abstract

Let $\Omega$ be an open set of $\mathbb{C}^n$ and $T$ be a positive closed current of dimension $p\geq 1$ on $\Omega$, we define a capacity associated to $T$ by: $C_T(K,\Omega)=C_T(K)={sup} \left\{\ds\int_K{T\wedge(dd^c v)^p,\ v\in {psh}(\Omega),\ 0<v<1}\right\}$ where $K$ is a compact set of $\Omega$. We prove, in the same way as Bedford-Taylor, that a locally bounded plurisubharmonic function is quasi-continuous with respect to $C_T$. In the second part we define the convergence relatively to $C_T$ and we prove that if $(u_j)$ is a family of locally uniformly bounded plurisubharmonic functions and $u$ is a locally bounded plurisubharmonic function such that $u_j \rightarrow u$ relatively to $C_T$ then $T\wedge (dd^cu_j)^p\rightarrow T\wedge (dd^cu)^p$ in the current sense.