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 $p\ge 1$ on $\Omega$, we define a capacity associated to $T$ by:

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 $u_j\to u$ relatively to $C_T$ then $T\wedge (d{d}^c{u}_j{)}^p\to T\wedge (d{d}^{c}u{)}^p$ in the current sense.