  A generalized transport theorem for convecting irregular domains is presented
in 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 open
set of an $n$-dimensional Euclidean space. The evolution of such a generalized
domain in time is assumed to be in accordance to a bi-Lipschitz type map. The
induced curve is shown to be continuous with respect to the flat norm and
differential with respect to the sharp norm on currents in $\mathbb{R}^{n}$. A
time dependent property is naturally assigned to the evolving region via the
action of an $r$-cochain on the current associated with the domain. Applying a
representation theorem for cochains the properties are shown to be locally
represented by an $r$-form. Using these notions a generalized transport theorem
is presented.
