I am learning current theory presently. In Chap 3.3-Definition of Monge-Ampère Operators, J.-P. Demailly, Complex analytic and differential geometry, I am a little confused as follows.
Let $X$ be a n-dimensional complex manifold. Let $u$ be a locally bounded plurisubharmonic function on $X$ and let $T$ be a closed positive current of bidimension $(p, p),$ i.e. of bidegree $(n-p, n-p) .$ Demailly defined the wedge $$ d d^{c} u \wedge T=d d^{c}(u T), $$ where $d d^{c}(\quad)$ is taken in the sense of distribution (or current) theory. Then he proved the following proposition.
Propostion The wedge product $d d^{c} u \wedge T$ is again a closed positive current.
Proof. The result is local. In an open set $\Omega \subset \mathbb{C}^{n},$ we can use convolution with a family of regularizing kernels to find a decreasing sequence of smooth plurisubharmonic functions $u_{k}=u \star \rho_{1 / k}$ converging pointwise to $u .$ Then $u \leqslant u_{k} \leqslant u_{1}$ and Lebesgue's dominated convergence theorem shows that $u_{k} T$ converges weakly to $u T ;$ thus $d d^{c}\left(u_{k} T\right)$.........
My question: In the above argument, Lebesgue's dominated convergence theorem is used. As we all know, Lebesgue's dominated convergence theorem reveals something about the integral. However, $T$ may be not the current induce by an $L_{\operatorname{loc}}^{1}$ function. So How can we apply Lebesgue's dominated convergence theorem to $u_{k} T$? Anyone know something similar with Lebesgue's dominated convergence theorem in the general current theory?
