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A generalized transport theorem for convecting irregular domains is presentedin the setting of Federer's geometric measure theory. A prototypical$r$-dimensional domain is viewed as a flat $r$-chain of finite mass in an openset of an $n$-dimensional Euclidean space. The evolution of such a generalizeddomain in time is assumed to be in accordance to a bi-Lipschitz type map. Theinduced curve is shown to be continuous with respect to the flat norm anddifferential with respect to the sharp norm on currents in $\mathbb{R}^{n}$. Atime dependent property is naturally assigned to the evolving region via theaction of an $r$-cochain on the current associated with the domain. Applying arepresentation theorem for cochains the properties are shown to be locallyrepresented by an $r$-form. Using these notions a generalized transport theoremis presented.

On the role of sharp chains in the transport theorem