We start with a few definitions.

It is clear by the definition that the closed positive currents $\gamma^{k}$, for $k=1,...,n$ are globally well defined. Consider now $\gamma\geq 0$ be a closed positive $(1,1)$-current with continuous local potentials. We define

Let $\varphi\in\hat{\cal P}_{\gamma}$ with zero Lelong numbers. It is well known from the first author work (which becomes drastically simple in this particular case) the existence of a family $(\varphi_{\varepsilon})_{\varepsilon>0}$, $\varphi_{\varepsilon}\in{\cal P}_{\gamma+\varepsilon\omega}\cap C^{\infty}(X)$, such that $\varphi_{\varepsilon}\downarrow\varphi$ as $\varepsilon\downarrow 0^{+}$. In the case $\varphi\in{\cal P}_{\gamma}\cap C^{0}(X)$ the convergence of $\varphi_{\varepsilon}$ is also uniform. We have the following crucial result.

As follows immediately from the proof, the statement of this theorem still holds if we replace $T^{l}$ with a product $T_{1}\wedge....\wedge T_{l}$, where the currents $T_{j}$ have the same properties as $T$. As a matter of fact, we wrote the statement in the previous special case only for the sake of notation simplicity. However during the proof it is useful to consider that statements concerning terms involving $T^{l}$ are still valid if we replace $T^{l}$ with $\gamma^{r}\wedge T^{l-r}$.

Proof. Statement (3.2) follows from (3.1) by using the weak continuity of the $i\partial\bar{\partial}$ operator and an induction on (3.2). We remark that claim 2 asserts statement A) in full generality for $k=0$. We denote by $A_{k,l}$ the assertion A) in the statement of the theorem for the relative indices $(k,l)$. For all $k=0,...,n-1$ and $l=0,...,n-k$ we define the following statement $B_{k,l}$: for all $p=0,...,k$

weakly as $\varepsilon\rightarrow 0^{+}$ and

We remark that (3.4) follows from (3.3) by the weak continuity of the $i\partial\bar{\partial}$ operator. By combining (3.4) with the weak continuity of the $i\partial\bar{\partial}$ operator we obtain

weakly as $\varepsilon\rightarrow 0^{+}$. On the other hand $\eqref{SymWeg}_{p-1,\bullet}$ implies

In this way we deduce (3.5). The symmetry identity (3.7) follows from (3.5) for $p=0$ and from the fact that

weakly as $\varepsilon\rightarrow 0^{+}$. This last convergence statement follows by combining (3.5) for $p=l=0$ with an induction on $l$ by means of the weak continuity of the $i\partial\bar{\partial}$ operator.
We now prove simultaneously the statements $A_{k,l}$ and $B_{k,l}$, $l=0,...,n-k$ by using an induction on $k=0,...,n-1$. For the moment we assume that the potential $\varphi$ in statement A) also has zero Lelong numbers, but we will get rid of this hypothesis at the end. Statements $A_{0,\bullet}$ and $B_{0,\bullet}$ are true by claim 2 and its proof. So we assume that these statements hold for $j\leq k-1$ and we prove them for $k$. The induction process is divided in two main steps.

Step I. This step consists in proving the

As pointed out before in order to prove $B_{k,l}$ is sufficient to show (3.3) and (3.6). The proof of (3.6) is quite similar to the proof of (3.3) that we now explain. We first prove by induction on $s=0,...,k-p$ the inequality

Inequality (3.8) is obviously true for $s=0$. (Here we adopt the usual convention of neglecting a sum when it runs over an empty set of indices.) Before procedding to the proof of inequality (3.8), we need to point out two useful remarks.

We prove this claim by induction on $l=0,...,n-k$. For $l=0$ the conclusion follows from the hypothesis $\varphi\in\hat{\cal P}_{\gamma}$. So we assume $A_{k,l-1}$ and we prove $A_{k,l}$. For this purpose set $T=\theta+i\partial\bar{\partial}u$, with $\theta$ smooth and $u$ continuous and expand the integral

Hypothesis $A_{k,l-1}$ implies by step I

weakly as $\varepsilon\rightarrow 0^{+}$. Thus by taking the limit as $\varepsilon\rightarrow 0^{+}$ in the previous identity and by combining the inductive hypothesis $\eqref{Mcv32}_{k-1,l-1}$ for $p=0$ with the weak continuity of the $i\partial\bar{\partial}$ operator, we deduce

By weak compactness of the mass we infer the existence of a sequence $(\varepsilon_{j})_{j}$, $\varepsilon_{j}\downarrow 0^{+}$ and a current of order zero $\Xi\in{\cal D}^{\prime}_{n-k-l,n-k-l}(X)$ such that

weakly as $j\rightarrow+\infty$. In particular

weakly as $j\rightarrow+\infty$. The fact that $\varphi_{\varepsilon_{j}}\downarrow\varphi$ and

weakly as $j\rightarrow+\infty$, by the convergence $(k-1)$-inductive hypothesis $\eqref{Mcv32}_{k-1,l}$, $p=0$ in the statement $B_{k-1,l}$, implies

thanks to lemma (3.9), page 189 in [Dem1]. We conclude

End of the proof. In the case the Lelong numbers of $\varphi$ are not zero we replace in the previous computations $\gamma$ with $\gamma+R\omega$ and $\varepsilon\omega$ with $0$. Here $R>0$ is chosen sufficiently big such that $0\leq\gamma+R\omega+i\partial\bar{\partial}\varphi_{\varepsilon}$ for all $\varepsilon\in(0,1)$ and $\varphi_{\varepsilon}\downarrow\varphi$ as $\varepsilon\rightarrow 0^{+}$. Then the previous arguments still work and statement A) rewrites as

Statement B) of the theorem rewrites as

weakly as $\varepsilon\rightarrow 0^{+}$ and $(\gamma_{\varphi}+R\omega)^{k}\wedge T^{l}=T^{l}\wedge(\gamma_{\varphi}+R\omega)^{k}$ for the relative indices $(k,l)$. The last inequality implies $\gamma_{\varphi}^{k}\wedge T^{l}=T^{l}\wedge\gamma_{\varphi}^{k}$. In fact this follows by expanding by linearity the term $(\gamma_{\varphi}+R\omega)^{k}$ and using an induction by means of formula (3.9). The base of the induction follows from claim 2. $\Box$

We consider also the subset $\check{\cal P}_{\gamma}:=\{\varphi\in\hat{\cal P}^{0}_{\gamma}\,\mid\,\int_{X}-\varphi\,\gamma_{\varphi}^{n}<+\infty\}+\mathbb{R}\subset\hat{\cal P}_{\gamma}$ . Without changes in the proof of theorem 5 we get the following corollary.

Let now $\Theta$ be a closed positive $(n-1,n-1)$-current and consider the $L^{2}$-space

equipped with the hermitian product $\left<\alpha,\beta\right>_{\Theta}:=\int_{X}i\alpha\wedge\bar{\beta}\wedge\Theta$, which is well defined by the polarization identity. The $\Theta$-almost everywhere equality relation is defined by : $\alpha\sim\beta$ iff

Let $\alpha_{k},\,\alpha\in L^{2}(X,\Theta)$. We say that the sequence $\alpha_{k}$ converges $L^{2}(X,\Theta)$-weakly to $\alpha$ if

for all $\beta\in L^{2}(X,\Theta)$. Let $\varphi\in{\cal P}^{0}_{\gamma}$ such that $\int_{X}-\varphi\,\Theta\wedge\,\omega<+\infty$. Then one can define $\partial\varphi\wedge\Theta:=\partial(\varphi\Theta)$. We write $\partial\varphi\in L^{2}(X,\Theta)$ if there exists $\alpha\in L^{2}(X,\Theta)$ such that $\partial(\varphi\Theta)=\alpha\wedge\Theta$ in the sense of currents. In this case we write

With this notations we have the following corolary of theorem 5.

Proof. By integrating by parts we obtain