  Given a smooth complex toric variety we will compare real Lagerberg forms and
currents on its tropicalization with invariant complex forms and currents on
the toric variety. Our main result is a correspondence theorem which identifies
the cone of invariant closed positive currents on the complex toric variety
with closed positive currents on the tropicalization. In a subsequent paper,
this correspondence will be used to develop a Bedford-Taylor theory of
plurisubharmonic functions on the tropicalization.
