In Demailly's e-book Complex analytic and differential geometry, chap3-(1.14) Proposition is stated as follows:
Every positive current $T=i^{(n-p)^{2}} \sum T_{I, J} d z_{I} \wedge d \bar{z}_{J}$ in $\mathscr{D}_{p, p}^{\prime+}(X)$ is real and of order 0, i.e. its coefficients $T_{I . J}$ are complex measures and satisfy $\overline{T_{I . J}}=T_{J, I}$ for all multi-indices $|I|=|J|=n-p .$ Moreover $T_{I, I} \geqslant 0,$ and the absolute values $\left|T_{I, J}\right|$ of the measures $T_{I, J}$ satisfy the inequality $$ \lambda_{I} \lambda_{J}\left|T_{I, J}\right| \leqslant 2^{p} \sum_{M} \lambda_{M}^{2} T_{M, M}, \quad I \cap J \subset M \subset I \cup J $$ where $\lambda_{k} \geqslant 0$ are arbitrary coefficients and $\lambda_{I}=\prod_{k \in I} \lambda_{k}$
My first question is :what is the definition of the absolute value of distribution $T_{I,J}$. Distribution is a global quantity , what is its pointwise value.
My 2-nd question is: how can we view the mass measure $\|T\|:=\sum\left|T_{I, J}\right|$ of a positive current as? What confuses me is the same reason as the first question(Distribution is a global quantity , what is its pointwise value.)
