I'm reading "complex analytic and differential geometry" by Demailly. In 3.5, before he define (Generalized) Lelong number, he assumes $X$ is a Stein manifold, $\varphi$ is a continuous plurisubharmonic function, and $T$ is a closed positive current of bidimension $(p,p)$. He further assumes that $\varphi$ is semi-exhaustive on $\operatorname{Supp} T$ (there is a real number $R$ such that $\{x\in X:\varphi(x)<R\}\cap\operatorname{Supp} T$ has a compact closure in $X$) and claims that the measure $T\wedge(dd^c\varphi)^p$ is well-defined because of the results in section 3.2.
Actually, I don't understand well why such measure is well-defined on the whole $X$ (this is well-defined on $X\setminus S(-\infty)$ by a result in the section 3.4, where $S(-\infty)$ is the $-\infty$-locus of $\varphi$) and in the section 3.2, I can find the theorem of El-Mir which says a closed positive current defined on $X\setminus E$ can be extended to the whole $X$ if the given current has finite mass locally on $E$ ($E$ is a complete pluripolar set). I doubt this can be applied to our situation since we don't know our $T$ has finite mass locally on $S(-\infty)$.
On the other hand, I want to know how the condition that $X$'s a Stein manifold can help us to define our measure well.
I guess I had an answer so let me try it.
Since $S(-\infty)\cap\operatorname{Supp} T$ is compact, there exists an $M$ such that $S(-\infty)\cap\operatorname{Supp} T\subset\{x:\psi(x)< M\}$, where $\psi$ is a smooth strongly plurisubharmonic function defining our Stein manifold. Note that $\{x:\psi(x)\leq M\}$ is compact and we can slightly shrink this to an open set with a smooth strongly pseudoconvex boundary which still includes $S(-\infty)\cap\operatorname{Supp} T$. So by section 4, our measure can be defined well. (In other words, our measure is locally finite everywhere.)