In Demailly's "Complex Analytic and Differential Geometry" page 20 2.D.4, he proves the Poincaré lemma for Currents. The theorem states as follows:
Let $\Omega\subset\mathbb R^m$ be a starshaped open subset,and $T$ is a closed current of degree $q$ and order $s$,then there exists a current $S$ of degree $q-1$ and of order $\leq s$,satisfying $dS=T$ on $\Omega$.
The most useful approach to prove this theorem is first show the fact that every closed current is cohomologous to a smooth form. (A)
In the proof of (A),he claims if $\Theta$ and all its derivatives are currents of order $0$, then $\Theta$ is smooth.
I know currents of order $0$ on manifold can be considered as differential forms with measure coefficients, but how can we get this claim?
Any suggestion and references are appreciated, thanks a lot!