A signed Radon measure $\mu$ on an open subset $W$ of ${X^{\rm an}}$ induces a current $[\mu]\in D_{0}(W)$ by setting $[\mu](f):=\int_{X^{\rm an}}fd\mu$ as usual. Since the topology on $A_{c}^{0}(W)=C_{c}^{\infty}(W)$ is finer than the topology induced by the supremum norm, we conclude that $[\mu]$ is indeed a current on $W$.