Proposition 1.2 . [02D9]

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Proposition 1.2.

Fix $\Omega$ a Kähler form on $X$ , and let $(\varepsilon_{j})$ be a sequence of positive real numbers decreasing to zero. Let $\varphi_{j}\in{\mathcal{E}}^{1}(X,\omega+\varepsilon_{j}\Omega)$ be a sequence of functions which decrease pointwise towards $\varphi$ , and such that

\sup_{j\geq 1}\int_{X}|\varphi_{j}|(\omega+\varepsilon_{j}\Omega+dd^{c}\varphi_{j})^{n}<+\infty.

Then $\varphi\in{\mathcal{E}}^{1}(X,\omega)$ , and $(\omega+\varepsilon_{j}\Omega+dd^{c}\varphi_{j})^{n}\rightarrow(\omega+dd^{c}\varphi)^{n}$ .