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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.

A comparison of positivity in complex and tropical toric geometry