Set $\omega_{j}:=\omega+\varepsilon_{j}\Omega$. We can assume w.l.o.g. that $\varphi_{j}\leq\varphi\leq 0$. Set

$$\varphi^{K}:=\max(\varphi,-K)\in PSH(X,\omega)\;\text{ and }\;\varphi_{j}^{K}:=\max(\varphi_{j},-K)\in PSH(X,\omega_{j}).$$

Observe that, $K$ being fixed, $(\varphi_{j}^{K})_{j}$ is uniformly bounded and decreases towards $\varphi^{K}$ as $j$ goes to infinity. Therefore $(\omega_{j}+dd^{c}\varphi_{j}^{K})^{n}\rightarrow(\omega+dd^{c}\varphi^{K})^{n}$, by a classical result of E.Bedford and A.Taylor [BT 82]. Moreover the sequence of positive measures $(-\varphi_{j}^{K})(\omega_{j}+dd^{c}\varphi_{j}^{K})^{n}$ has uniformly bounded mass, since by lemma 7.2 in [GZ 2],

$$0\leq\int_{X}(-\varphi_{j}^{K})(\omega_{j}+dd^{c}\varphi_{j}^{K})^{n}\leq 2^{n}\int_{X}(-\varphi_{j})(\omega_{j}+dd^{c}\varphi_{j})^{n}\leq 2^{n}M,$$

where $M:=\sup_{j}\int_{X}(-\varphi_{j})(\omega_{j}+dd^{c}\varphi_{j})^{n}<+\infty$.

Since $\varphi_{j}$ is u.s.c., a standard argument yields that any cluster point $\nu$ of the sequence $(-\varphi_{j}^{K})(\omega_{j}+dd^{c}\varphi_{j}^{K})^{n}$ satisfies $0\leq(-\varphi^{K})(\omega+dd^{c}\varphi^{K})^{n}\leq\nu$. In particular

$$0\leq\int_{X}(-\varphi^{K})(\omega+dd^{c}\varphi^{K})^{n}\leq\liminf_{j\rightarrow+\infty}\int(-\varphi_{j}^{K})(\omega_{j}+dd^{c}\varphi_{j}^{K})^{n}\leq 2^{n}M$$

is bounded from above uniformly with respect to $K$. Since $\varphi^{K}$ decreases towards $\varphi$, this shows $\varphi\in{\mathcal{E}}^{1}(X,\omega)$.

It remains to show that $(\omega_{j}+dd^{c}\varphi_{j})^{n}\rightarrow(\omega+dd^{c}\varphi)^{n}$. Since $(\omega_{j}+dd^{c}\varphi_{j}^{K})^{n}\rightarrow(\omega+dd^{c}\varphi^{K})^{n}$ for any fixed $K$, it is enough to get an upper bound on the mass of $(\omega_{j}+dd^{c}\varphi_{j}^{K})^{n}$ in $(\varphi_{j}\leq-K)$ which is uniform in $j$. This follows from Chebyshev inequality, namely

$$\int_{(\varphi_{j}\leq-K)}(\omega_{j}+dd^{c}\varphi_{j})^{n}\leq\frac{1}{K}\int_{X}(-\varphi_{j})(\omega_{j}+dd^{c}\varphi_{j})^{n}\leq\frac{M}{K}.$$

This yields the desired result. ∎