(Degenerate monotone convergence result).
Let $(X,\omega)$ be a polarized compact Kähler manifold of complex dimension $n$ and let $\gamma$, $T$ be closed positive $(1,1)$-currents with continuous local potentials. Then the following statements hold true.
A) For all $\varphi\in\hat{\cal P}_{\gamma}$, $\varphi\leq 0$ and $k,l\geq 0$, $k+l\leq n$, $k\leq n-1$

$$\int\limits_{X}-\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\wedge\omega^{n-k-l}<+\infty\,,$$

B) Let $\varphi\in\hat{\cal P}_{\gamma}$, $\varphi\leq 0$ with zero Lelong numbers and $\varphi_{\varepsilon}\in{\cal P}_{\gamma+\varepsilon\omega}\cap C^{\infty}(X)$, such that $\varphi_{\varepsilon}\downarrow\varphi$ as $\varepsilon\rightarrow 0^{+}$. Then for all $k,l\geq 0$, $k+l\leq n$, $k\leq n-1$

	$\displaystyle\varphi_{\varepsilon}\,(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k}\wedge T^{l}\longrightarrow\varphi\,\gamma_{\varphi}^{k}\wedge T^{l}\,,$		(3.1)
	$\displaystyle(\gamma_{\varphi_{\varepsilon}}+\varepsilon\omega)^{k+1}\wedge T^{l}\longrightarrow\gamma_{\varphi}^{k+1}\wedge T^{l}\,,$		(3.2)

weakly as $\varepsilon\rightarrow 0^{+}$. Moreover $\gamma_{\varphi}^{k}\wedge T^{l}=T^{l}\wedge\gamma_{\varphi}^{k}$ for all $\varphi\in\hat{\cal P}_{\gamma}$ and $k,l\geq 0$, $k+l\leq n$.