An $m$ -dimensional rectifiable current is a current $T$ whose action against a $m$ -form $\omega$ can be written as $$ T(\omega) = \int_S \theta(x)\langle \xi(x),\omega(x)\rangle d\mathcal H^m(x) $$ where $S$ is an $m$ -dimensional bounded rectifiable set, $\xi$ is an orientation of $S$ i.e. $\xi(x)$ is a unit $m$ -vector representing the approximate tangent plane of $S$ at $x$ for $\mathcal H^m$ -a.e. $x\in S$ and, finally, $\theta(x)$ is an integer valued measurable function defined a.e. on $S$ (called multiplicity). The space of $m$ -dimensional rectifiable currents is denoted by $\mathcal R_m$ .

An $m$ -dimensional rectifiable current $T$ such that the boundary $\partial T$ is itself an $(m-1)$ -dimensional rectifiable current, is called integral current. The space of integral currents is denoted by $\mathbf I_m$ . We point out that the word ``integral'' refers to the fact that the multiplicity $\theta$ is integer valued.

Also notice that rectifiable and integral currents are not vector subspaces of the space of currents. In fact while the sum of two rectifiable currents is again a rectifiable current, the multiplication by a real number gives a rectifiable current only if the number is an integer.

The compactness theorem makes the space of integral currents a good space where geometric problems can be ambiented.

On rectifiable currents one can define an integral flat norm $$ \mathcal F(T) := \inf \{\mathbf M(A) + \mathbf M(B)\colon T=A+\partial B,\quad A\in \mathcal R_m,\ B\in\mathcal R_{m+1}\}. $$

The closure of the space $\mathcal R_m$ under the integral flat norm is called the space of integral flat chains and is denoted by $\mathcal F_m$ .

As a consequence of the closure theorem, one finds that $\mathcal R_m = \{T\in \mathcal F_m\colon \mathbf M(T)<\infty\}$ where $\mathbf M$ is the mass norm of a current.