Integral current

Remarks

By the boundary rectifiability theorem of Federer and Fleming, the boundary of an integral current is again an integral current. Federer's original definition asks instead that $T$ and $\partial T$ be rectifiable currents, and the two definitions agree by that theorem. The integral $0$-currents are the finite sums $\sum_i a_i \delta_{p_i}$ with $a_i \in \Z$.

Example: let $D$ be the closed unit disc in $\R^2 \subseteq \R^3$, oriented by $e_1 \wedge e_2$. Then $[\![D]\!](\omega) = \int_D \omega$ is an integral $2$-current: take $S = D$, $\theta = 1$, $\xi = e_1 \wedge e_2$. Its mass is $\pi$, and its boundary is the circle, of mass $2\pi$.

Non-example: in $\R$, let $T = \sum_{k \ge 1} [\![I_k]\!]$, where $I_k = [1/(2k + 1), 1/(2k)]$ is oriented by $e_1$. It is integer multiplicity rectifiable with compact support in $[0, 1]$ and $\mathbf{M}(T) = \sum_k \frac{1}{2k(2k + 1)} < \infty$. But $\partial T$ is a sum of infinitely many unit point masses, so $\mathbf{M}(\partial T) = \infty$ and $T$ is not an integral current.