Let $X$ be a compact complex manifold of complex dimension $n$ and let $\gamma$ be a closed real $(1,1)$-current with continuous local potentials or a closed positive $(1,1)$-current with bounded local potentials. Then to any distribution $\Psi$ on $X$ such that $\gamma+i\partial\bar{\partial}\Psi\geq 0$ we can associate a unique locally integrable and bounded from above function $\psi:X\rightarrow[-\infty,+\infty)$ such that the corresponding distribution coincides with $\Psi$ and such that for any continuous or plurisubharmonic local potential $h$ of $\gamma$ the function $h+\psi$ is plurisubharmonic. In fact let $\Omega_{\alpha}$, $\Omega_{\beta}\subset X$ two open sets such that $\gamma=i\partial\bar{\partial}h_{\alpha}$ over $\Omega_{\alpha}$, $\gamma=i\partial\bar{\partial}h_{\beta}$ over $\Omega_{\beta}$, with $h_{\alpha}\in C^{0}(\Omega_{\alpha})$, $h_{\beta}\in C^{0}(\Omega_{\beta})$ in the first case or $h_{\alpha}\in\operatorname{Psh}(\Omega_{\alpha})$, $h_{\beta}\in\operatorname{Psh}(\Omega_{\beta})$ in the second case. Consider now the distributions $U_{\alpha}:=h_{\alpha}+\Psi$ and $U_{\beta}:=h_{\beta}+\Psi$. Then the hypothesis $\gamma+i\partial\bar{\partial}\Psi\geq 0$ implies the existence of uniques $u_{\alpha}\in\operatorname{Psh}(\Omega_{\alpha})$, $u_{\beta}\in\operatorname{Psh}(\Omega_{\beta})$ such that the corresponding distributions are respectively $U_{\alpha}$ and $U_{\beta}$. We set $\psi_{\alpha}:=u_{\alpha}-h_{\alpha}$ and $\psi_{\beta}:=u_{\beta}-h_{\beta}$. The function $h_{\beta}-h_{\alpha}$ is harmonic in both cases. Then the identity $U_{\beta}=U_{\alpha}+h_{\beta}-h_{\alpha}$ over $\Omega_{\alpha}\cap\Omega_{\beta}$ implies that the plurisubharmonic functions $u_{\beta}$ and $u_{\alpha}+h_{\beta}-h_{\alpha}$ represent the same distribution so they coincide over $\Omega_{\alpha}\cap\Omega_{\beta}$. We deduce $\psi_{\alpha}=\psi_{\beta}$ over $\Omega_{\alpha}\cap\Omega_{\beta}$. In this way we obtain a global function $\psi:X\rightarrow[-\infty,+\infty)$ which satisfies the required properties since the function $h-h_{\alpha}$ is harmonic in both cases. Then the uniqueness of $\psi$ is obvious. The set of functions $\psi$ obtained in this way will be denoted by ${\cal P}_{\gamma}$. We set ${\cal P}^{0}_{\gamma}:=\{\psi\in{\cal P}_{\gamma}\,\mid\,\sup_{X}\psi=0\}$. A closed positive $(1,1)$-current with bounded local potentials such that $\{\gamma\}^{n}:=\int_{X}\gamma^{n}>0$, will be called big.

Statement (C) will follow from the arguments of the proof of theorem 3.

Remark 1. As application of his estimates, Kołodziej considers in Example 1, page 91 of [Kol] Monge-Ampère equations with non degenerate left hand side and with right hand side taking values in the Orlicz space $L^{\Psi}(X)$, with $\Psi(t):=t\log^{n}(e+t)\log^{n+\delta}(e+\log(1+t))$, $\delta>0$. If we take $\varepsilon_{0}=1/k$ with an integer $k>1$ we obtain

$$\lim_{t\rightarrow+\infty}\frac{\log^{n+\delta}(e+\log(1+t))}{\log^{\varepsilon_{0}}(e+t)}=+\infty\,.$$

This implies $L^{\Psi}(X)\subset L\log^{n+\varepsilon_{0}}L(X)$.

Let $X$ be a compact complex manifold of complex dimension $n$, let $\gamma$ be a big closed positive $(1,1)$-current with bounded local potentials. Set ${\cal P}_{\gamma}[0,1]:=\{\varphi\in{\cal P}_{\gamma}\,\mid\,0\leq\varphi\leq 1\}$ and

$$\operatorname{Cap}_{\gamma}(E):=\sup_{\varphi\in{\cal P}_{\gamma}[0,1]}\,\{\gamma\}^{-n}\int\limits_{E}\gamma^{n}_{\varphi}\,,$$

for all Borel sets $E\subset X$. We remark that if $(E_{j})_{j}$, $E_{j}\subset E_{j+1}\subset X$ is a family of Borel sets and $E=\bigcup_{j}E_{j}$ then

$\displaystyle\operatorname{Cap}_{\gamma}(E)=\lim_{j\rightarrow+\infty}\operatorname{Cap}_{\gamma}(E_{j})\,.$

(2.1)

In fact for all $\varepsilon>0$ let $\varphi\in{\cal P}_{\gamma}[0,1]$ such that $\{\gamma\}^{-n}\int_{E}\gamma^{n}_{\varphi}>\operatorname{Cap}_{\gamma}(E)-\varepsilon$. Then

	$\displaystyle\operatorname{Cap}_{\gamma}(E)$	$\displaystyle\geq$	$\displaystyle\lim_{j\rightarrow+\infty}\operatorname{Cap}_{\gamma}(E_{j})$
		$\displaystyle\geq$	$\displaystyle\lim_{j\rightarrow+\infty}\,\{\gamma\}^{-n}\int\limits_{E_{j}}\gamma^{n}_{\varphi}=\,\{\gamma\}^{-n}\int\limits_{E}\gamma^{n}_{\varphi}>\operatorname{Cap}_{\gamma}(E)-\varepsilon\,.$

The first two integral estimates of lemma (1) are quite standard in the elementary theory of plurisubharmonic functions and the dependence of the constants $\alpha$ and $C$ on $\gamma$ is only on the $L^{\infty}$ bound of its local potentials. To be more precise concerning the uniform estimate $\int_{X}e^{-\alpha\psi}\,\Omega\leq C$ one can make the constant $\alpha$ depending only on the cohomology class of $\gamma$ as in [Ti1], but in this case the constant $C$ will depend on the $L^{\infty}$ bound of the local potentials of $\gamma$ and on the volume form $\Omega$. One can also make $C$ depending only on the volume form $\Omega$, but in this case $\alpha$ will depend on the $L^{\infty}$ bound of the local potentials of $\gamma$ and on the volume form $\Omega$.

Proof. We remark first the inequality

$$\int\limits_{\psi<-t}\gamma^{n}_{\varphi}\;\leq\frac{1}{t}\int\limits_{X}-\psi\,\gamma^{n}_{\varphi}\,,$$

and we prove the following elementary claim.

Proof. The fact that the current $\gamma$ is positive implies $\psi_{c}:=\max\{\psi,c\}\in{\cal P}_{\gamma}$, $c\in\mathbb{R}_{<0}$, so by the monotone convergence theorem it is sufficent to prove the inequality (2.2) for $\psi\in{\cal P}_{\gamma}\cap L^{\infty}(X)$. So assume this and let $\omega>0$ be a hermitian metric over $X$. By a result of Greene-Wu [Gr-Wu] there exist a family of functions $(\psi_{\varepsilon})_{\varepsilon>0}$, $\psi_{\varepsilon}\in{\cal P}_{\gamma+\varepsilon\omega}\cap C^{\infty}(X)$ such that $\psi_{\varepsilon}\downarrow\psi$ as $\varepsilon\rightarrow 0^{+}$. Consider now the integrals $I_{j}:=\int_{X}-\psi\,\gamma^{j}\wedge\gamma_{\varphi}^{n-j}\,$ for all $j=0,...,n$. Then $I_{j}\leq I_{j+1}+\int_{X}\gamma^{n}$. In fact by Stokes formula

	$\displaystyle I_{j}$	$\displaystyle=$	$\displaystyle I_{j+1}-\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}\psi_{\varepsilon}\,\gamma^{j}\wedge i\partial\bar{\partial}\varphi\wedge\gamma_{\varphi}^{n-j-1}$
		$\displaystyle=$	$\displaystyle I_{j+1}-\lim_{\varepsilon\rightarrow 0^{+}}\,\int\limits_{X}i\partial\bar{\partial}\psi_{\varepsilon}\wedge\varphi\,\gamma^{j}\wedge\gamma_{\varphi}^{n-j-1}$
		$\displaystyle\leq$	$\displaystyle I_{j+1}+\int\limits_{X}\varphi\,\gamma^{j+1}\wedge\gamma_{\varphi}^{n-j-1}\leq I_{j+1}+\int\limits_{X}\gamma^{n}\,.$

In this way we deduce the required inequality $I_{0}\leq I_{n}+n\int_{X}\gamma^{n}$. $\Box$

The following claim will be very useful for the rest of the paper.

Proof. The proof of the convergence of the constants $C_{l}$ goes by induction on $l=0,...,n$. The statement is true for $l=0$ by the first integral estimate of lemma 1. So we assume it is true for $l$ and prove it for $l+1$. Let $\psi_{c}:=\max\{\psi,c\}\in{\cal P}_{\gamma}$, $c\in\mathbb{R}_{<0}$. The inductive hypothesis allows us to apply the decreasing monotone convergence theorem in order to deduce

$$\lim_{c\rightarrow-\infty}\,\int\limits_{X}(\psi_{c}-\psi)T^{l}\wedge\omega^{n-l}=0\,,$$

which means the converegence in mass $\|(\psi_{c}-\psi)T^{l}\|_{\omega}(X)\rightarrow 0$ as $c\rightarrow-\infty$, in particular $\psi_{c}T^{l}\rightarrow\psi T$ weakly as $c\rightarrow-\infty$. So by the weak continuity of the $i\partial\bar{\partial}$ operator we deduce $i\partial\bar{\partial}\psi_{c}\wedge T^{l}\rightarrow i\partial\bar{\partial}\psi\wedge T$ and

$\displaystyle\gamma_{\psi_{c}}\wedge T^{l}\longrightarrow\gamma_{\psi}\wedge T^{l}\,,$

(2.3)

weakly as $c\rightarrow-\infty$. By the result of Greene-Wu [Gr-Wu] let $(\psi_{c,\varepsilon})_{\varepsilon>0}$, $\psi_{c,\varepsilon}\in{\cal P}_{\gamma+\varepsilon\omega}\cap C^{\infty}(X)$ such that $\psi_{c,\varepsilon}\downarrow\psi_{c}$ as $\varepsilon\rightarrow 0^{+}$ and write $T=\theta+i\partial\bar{\partial}u$, whith $\theta$ smooth, $\theta\leq K\omega$ and $u$ continuous whith $\min_{X}u=0$. By using the monotone convergence theorem, Stokes formula and (2.3), we expand the integral

			$\displaystyle\int\limits_{X}-\psi\,T^{l+1}\wedge\omega^{n-l-1}=\lim_{c\rightarrow-\infty}\;\lim_{\varepsilon\rightarrow 0^{+}}\;\int\limits_{X}-\psi_{c,\varepsilon}\,T^{l+1}\wedge\omega^{n-l-1}$
		$\displaystyle=$	$\displaystyle\lim_{c\rightarrow-\infty}\;\lim_{\varepsilon\rightarrow 0^{+}}\,\left[\;\int\limits_{X}-\psi_{c,\varepsilon}\,\theta\wedge T^{l}\wedge\omega^{n-l-1}-\int\limits_{X}\psi_{c,\varepsilon}\,i\partial\bar{\partial}u\wedge T^{l}\wedge\omega^{n-l-1}\right]$
		$\displaystyle\leq$	$\displaystyle\lim_{c\rightarrow-\infty}\;\lim_{\varepsilon\rightarrow 0^{+}}\,\left[\;\int\limits_{X}-\psi_{c,\varepsilon}\,T^{l}\wedge K\omega^{n-l}-\int\limits_{X}u\,i\partial\bar{\partial}\psi_{c,\varepsilon}\wedge T^{l}\wedge\omega^{n-l-1}\right]$
		$\displaystyle=$	$\displaystyle\int\limits_{X}-\psi\,T^{l}\wedge K\omega^{n-l}-\int\limits_{X}u\,i\partial\bar{\partial}\psi\wedge T^{l}\wedge\omega^{n-l-1}$
		$\displaystyle\leq$	$\displaystyle KC_{l}-\int\limits_{X}u\,\gamma_{\psi}\wedge T^{l}\wedge\omega^{n-l-1}+\int\limits_{X}u\,\gamma\wedge T^{l}\wedge\omega^{n-l-1}$
		$\displaystyle\leq$	$\displaystyle KC_{l}+\max_{X}u\,\int\limits_{X}\gamma\wedge T^{l}\wedge\omega^{n-l-1}<+\infty\,,$

by the inductive hypothesis. Concerning the symmetry of the exterior product we remark that the weak continuity of the $i\partial\bar{\partial}$ operator implies by induction on $l$

$$T^{l}\wedge\gamma_{\psi_{c}}\longrightarrow T^{l}\wedge\gamma_{\psi}\,,$$

weakly as $c\rightarrow-\infty$. This combined with (2.3) implies $\gamma_{\psi}\wedge T^{l}=T^{l}\wedge\gamma_{\psi}$ . $\Box$

In the particular case $T=\gamma$ the constant

$$0<C(\gamma):=\sup_{\psi\in{\cal P}^{0}_{\gamma}}\;\;\{\gamma\}^{-n}\int\limits_{X}-\psi\,\gamma^{n}<+\infty$$

satisfies the capacity estimate of the statement (A) of the lemma 2, by the claim 1. Moreover the previous induction give us

$$C_{1}\leq K\int\limits_{X}-\psi\,\omega^{n}+\max_{X}u\,\int\limits_{X}\gamma\wedge\omega^{n-1}\leq K\int\limits_{X}-\psi\,\omega^{n}+RK\,\int\limits_{X}\omega^{n}\,,$$

where $R\geq\max_{X}u$ and in general

$$C_{l+1}\leq KC_{l}+R\int\limits_{X}\gamma^{l+1}\wedge\omega^{n-l-1}\leq KC_{l}+RK^{l+1}\,\int\limits_{X}\omega^{n}\,.$$

We deduce $C_{n}\leq K^{n}\int_{X}-\psi\,\omega^{n}+nRK^{n}\int_{X}\omega^{n}$. Thus if $(\gamma_{t})_{t>0}$ is a family satisfying the hypothesis $(C1)$ and $(C2a)$ of the statement $(C)$ in theorem 3 and $K_{t}=(\{\gamma_{t}\}^{n})^{1/n}$ then the constant $C(\gamma)$ satisfies the stability properties of the statement (A) of the lemma 2. We prove now the statement (B) of the lemma 2. In fact let $f:=\{\gamma\}^{-n}\gamma^{n}/\Omega\geq 0$. Then the uniform estimate for the integral

$$\{\gamma\}^{-n}\int\limits_{X}-\psi\,\gamma^{n}=\frac{1}{\alpha}\,\int\limits_{X}-\alpha\psi f\,\Omega$$

follows from the elementary inequality $-\alpha\psi f\leq e^{-\alpha\psi}-1+f\log(1+f)$ combined with the uniform estimate $\int_{X}e^{-\alpha\psi}\Omega\leq C$ of lemma 1. In this case the required stability properties of the constant $C(\gamma,\Omega)>0$ in the capacity estimate are obvious. $\Box$

Proof.
Step I. We assume first $\varphi,\,\psi\in{\cal P}_{\gamma}\cap C^{0}(X)$. We will denote by $\partial S$ the boundary in $X$ of a set $S\subset X$. By the continuity of the functions $\varphi,\,\psi$ we deduce:
1) the set $\{\varphi<\psi\}$ is open and $\partial\{\varphi<\psi\}\subset\{\varphi=\psi\}$,
2) for all $\varepsilon>0$ there exists an open neighborhood ${\cal V}\subset X$ of the set $\{\varphi\geq\psi\}$ such that $\max\{\varphi+\varepsilon,\psi\}=\varphi+\varepsilon$ over ${\cal V}$.
So $\partial\{\varphi<\psi\}\subset{\cal V}$ and the Stokes formula implies the equality

$\displaystyle\int\limits_{\varphi<\psi}\gamma^{n}_{\varphi}\;=\int\limits_{\varphi<\psi}(\gamma+i\partial\bar{\partial}\max\{\varphi+\varepsilon,\psi\})^{n}\,,$

for all $\varepsilon>0$. Moreover by the monotone convergence theorem in pluripotential theory we deuce that the current $(\gamma+i\partial\bar{\partial}\max\{\varphi+\varepsilon,\psi\})^{n}$ converges weakly to the current $\gamma_{\psi}^{n}$ over the open set $\{\varphi<\psi\}$ as $\varepsilon\rightarrow 0^{+}$. Thus

$\displaystyle\int\limits_{\varphi<\psi}\gamma_{\varphi}^{n}\;=\,\liminf\limits_{\varepsilon\rightarrow 0^{+}}\int\limits_{\varphi<\psi}(\gamma+i\partial\bar{\partial}\max\{\varphi+\varepsilon,\psi\})^{n}\geq\int\limits_{\varphi<\psi}\gamma_{\psi}^{n}\,.$

Step II. For the general case we use the following well known fact.

Let now $(U_{\alpha})_{\alpha=1}^{N}$ be a finite open covering of $X$ such that $\gamma=i\partial\bar{\partial}h_{\alpha}$ over $U_{\alpha}$ with $\inf_{U_{\alpha}}h_{\alpha}=0$. By the quasicontinuity of plurisubharmonic functions, for every $\delta>0$ there exists an open set $G_{\delta,\alpha}\subset U_{\alpha}$ such that $u_{\alpha}:=h_{\alpha}+\varphi,\,h_{\alpha}\in C^{0}(U_{\alpha}\smallsetminus G_{\delta,\alpha})$ and $\operatorname{Cap}(G_{\delta,\alpha},U_{\alpha})<\delta$. In particular $\varphi\in C^{0}(U_{\alpha}\smallsetminus G_{\delta,\alpha})$. Consider the open set $G_{\delta}:=\bigcup_{\alpha}G_{\delta,\alpha}\subset X$. Then the inclusion $X\smallsetminus G_{\delta}\subset\bigcup_{\alpha}(U_{\alpha}\smallsetminus G_{\delta,\alpha})$ implies $\varphi\in C^{0}(X\smallsetminus G_{\delta})$ by claim 3. We can also assume $\psi\in C^{0}(X\smallsetminus G_{\delta})$. Set also $v_{\alpha}:=h_{\alpha}+\psi$. Let $\omega$ be a Kähler metric over $X$ and $g_{\alpha}$ smooth functions over $U_{\alpha}$ such that $\omega=i\partial\bar{\partial}g_{\alpha}$, $\inf_{U_{\alpha}}g_{\alpha}=0$. By the result of Greene-Wu [Gr-Wu] there exists a sequence $(\varepsilon_{j})_{j}\subset(0,\varepsilon)$, $\varepsilon_{j}\downarrow 0$ and $\psi_{j},\,\varphi_{j}\in{\cal P}_{\gamma+\varepsilon_{j}\omega}\cap C^{\infty}(X)$, with $\psi_{j}\downarrow\psi$ and $\varphi_{j}\downarrow\varphi$. We can assume $0\leq\inf_{X}\varphi$, $0\leq\inf_{X}\psi$ and we set $u_{j,\alpha}:=h_{\alpha}+g_{\alpha}+\varphi_{j}$, $v_{j,\alpha}:=h_{\alpha}+g_{\alpha}+\psi_{j}$, $M:=\max_{\alpha}\{\|u_{1,\alpha}\|_{L^{\infty}(U_{\alpha})},\|v_{1,\alpha}\|_{L^{\infty}(U_{\alpha})}\}$. Then by step I

$\displaystyle\int\limits_{\varphi_{k}<\psi_{j}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\psi_{j})^{n}\leq\int\limits_{\varphi_{k}<\psi_{j}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}\;.$

(2.4)

Let $f\in C^{0}(X)$ such that $f=\psi$ over $X\smallsetminus G_{\delta}$. Then the set $\{\varphi_{k}<f\}$ is open and $\{\varphi_{k}<f\}\cup G_{\delta}=\{\varphi_{k}<\psi\}\cup G_{\delta}$. Thus

			$\displaystyle\int\limits_{\varphi_{k}<\psi}\gamma_{\psi}^{n}\;\leq\int\limits_{\varphi_{k}<f}\gamma_{\psi}^{n}+\int\limits_{G_{\delta}}\gamma_{\psi}^{n}\,\leq\int\limits_{\varphi_{k}<f}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\psi)^{n}+\sum_{\alpha}\;\int\limits_{G_{\delta,\alpha}}(i\partial\bar{\partial}v_{\alpha})^{n}$
		$\displaystyle\leq$	$\displaystyle\liminf_{j\rightarrow+\infty}\int\limits_{\varphi_{k}<f}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\psi_{j})^{n}+M^{n}\sum_{\alpha}\;\int\limits_{G_{\delta,\alpha}}(i\partial\bar{\partial}M^{-1}\,v_{\alpha})^{n}$
		$\displaystyle\leq$	$\displaystyle\liminf_{j\rightarrow+\infty}\left(\;\int\limits_{\varphi_{k}<\psi_{j}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\psi_{j})^{n}+\int\limits_{G_{\delta}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\psi_{j})^{n}\right)+M^{n}N\delta$
		$\displaystyle\leq$	$\displaystyle\liminf_{j\rightarrow+\infty}\left(\;\int\limits_{\varphi_{k}<\psi_{j}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}+M^{n}\sum_{\alpha}\;\int\limits_{G_{\delta,\alpha}}(i\partial\bar{\partial}M^{-1}\,v_{j,\alpha})^{n}\right)$
		$\displaystyle+$	$\displaystyle M^{n}N\delta\qquad\qquad\mbox{(by \eqref{CompPrincC0}) }$
		$\displaystyle\leq$	$\displaystyle\lim_{j\rightarrow+\infty}\int\limits_{\varphi_{k}<\psi_{j}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}+2M^{n}N\delta$
		$\displaystyle=$	$\displaystyle\int\limits_{\varphi_{k}\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}+2M^{n}N\delta\,.$

Then by letting $k\rightarrow+\infty$ we get

$\displaystyle\int\limits_{\varphi<\psi}\gamma_{\psi}^{n}\;\leq\,\limsup_{k\rightarrow+\infty}\int\limits_{\varphi_{k}\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}+2M^{n}N\delta\,.$

(2.5)

Now the set $\{\varphi\leq\psi\}\smallsetminus G_{\delta}$ is closed by the continuity of $\varphi$ and $\psi$ over $X\smallsetminus G_{\delta}$. Thus

			$\displaystyle\int\limits_{\varphi\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi)^{n}\geq\int\limits_{\{\varphi\leq\psi\}\smallsetminus G_{\delta}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi)^{n}$
		$\displaystyle\geq$	$\displaystyle\limsup_{k\rightarrow+\infty}\int\limits_{\{\varphi\leq\psi\}\smallsetminus G_{\delta}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}$
		$\displaystyle\geq$	$\displaystyle\limsup_{k\rightarrow+\infty}\left(\;\,\int\limits_{\varphi\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}-\int\limits_{G_{\delta}}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}\right)$
		$\displaystyle\geq$	$\displaystyle\limsup_{k\rightarrow+\infty}\left(\;\,\int\limits_{\varphi\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}-M^{n}\sum_{\alpha}\;\int\limits_{G_{\delta,\alpha}}(i\partial\bar{\partial}M^{-1}\,u_{k,\alpha})^{n}\right)$
		$\displaystyle\geq$	$\displaystyle\limsup_{k\rightarrow+\infty}\int\limits_{\varphi_{k}\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi_{k})^{n}-M^{n}N\delta\,.$

So by (2.5) we derive

	$\displaystyle\int\limits_{\varphi<\psi}\gamma_{\psi}^{n}\;$	$\displaystyle\leq$	$\displaystyle\int\limits_{\varphi\leq\psi}(\gamma+\varepsilon\omega+i\partial\bar{\partial}\varphi)^{n}+3M^{n}N\delta$
		$\displaystyle\leq$	$\displaystyle\int\limits_{\varphi\leq\psi}\gamma_{\varphi}^{n}\;+\;\sum_{l=1}^{n}{n\choose l}\int\limits_{X}\varepsilon^{l}\omega^{l}\wedge\gamma^{n-l}+3M^{n}N\delta\,.$

Then letting $\varepsilon\rightarrow 0$ and $\delta\rightarrow 0$ we get

$$\int\limits_{\varphi<\psi}\gamma_{\psi}^{n}\;\leq\int\limits_{\varphi\leq\psi}\gamma_{\varphi}^{n}\;.$$

Now the conclusion follows by replacing $\varphi$ by $\varphi+t$, $t>0$ in the previous formula and letting $t\rightarrow 0$. $\Box$

We recall now the following lemma due to Kołodziej [Kol], (see also [Ti-Zhu1], [Ti-Zhu2]).

The following lemma is a simple application of the main result in Bedford-Taylor [Be-Te].

Proof. In order to prove this estimate, it is sufficient by (2.1) to show the inequality

$\displaystyle\int\limits_{U}\Omega\leq e^{\alpha}Ce^{-\alpha/\operatorname{Cap}_{\gamma}(\,\overline{U}\,)^{1/n}}$

for an arbitrary relatively compact open set $U\subset\subset V$. For this purpose, consider the function

$$\Psi_{U}(x):=\sup\{\varphi(x)\,\mid\,\varphi\in{\cal P}_{\gamma}\,,\,\varphi_{\mid_{U}}\leq 0\}\geq 0\,.$$

Remark that $(\Psi_{U})_{\mid_{U}}\equiv 0$ since $0\in{\cal P}_{\gamma}$. If $U\not=\emptyset$ there exists a constant $C_{U}>0$ such that $\sup_{X}\,\varphi\leq C_{U}$ for all $\varphi\in{\cal P}_{\gamma}\,,\,\varphi_{\mid_{U}}\leq 0$. In fact let $S_{U}:=\{\varphi\in{\cal P}_{\gamma}\,\mid\,\varphi_{\mid_{U}}\leq 0\}$ and set $\tilde{\varphi}:=\varphi-\sup_{X}\varphi$. By contradiction we would get $\varphi_{j}\in S_{U}$ such that $\sup_{X}\varphi_{j}\rightarrow+\infty$. This implies $\sup_{U}\tilde{\varphi}_{j}\rightarrow-\infty$ and so $\int_{U}-\tilde{\varphi}_{j}\,\Omega\geq-C\sup_{U}\tilde{\varphi}_{j}\rightarrow+\infty$, which contradicts the first integral estimate of lemma 1.
Then it follows from quite standard local arguments that the upper regularization $\Psi^{*}_{U}\in{\cal P}_{\gamma}$. Moreover $\Psi^{*}_{U}\in L^{\infty}(X)$, $\Psi^{*}_{U}\geq 0$ and $\Psi^{*}_{U}\equiv 0$ over $U$. We recall now the following well known consequence of a result of Bedford and Taylor [Be-Te].