is divergence free and by Theorem \[teo:federer\], $\sigma =(u^+-u^-)\nu_u^\perp \H^1=U(J_u,[u],\nu_u^\perp)$.

Let us now produce an elementary example of measure $\g$ of the form $U(M,\t,\xi)$.

\[ex:div2punti\] Given two points $x,y\in{\Omega}$ we consider the smooth path from $x$ to $y$ defined as $$r(t):=x+t\frac{y-x}{|x-y|}\quad \mbox{ for } t\in [0,|x-y|].$$ We define the measure $\g\in \M({\Omega},{\mathbf{R}}^2)$ by $$(\phi,\g)\,:=\,\int\langle\phi(r(t)),\dot{r}(t)\rangle\dt\qquad\mbox{for any } \phi\in \C(\overline{{\Omega}},{\mathbf{R}}^2).$$ We then have $\g=U([x,y],1,\xi)$ with $\xi=\frac{y-x}{|y-x|}$. Notice that ${\nabla}\cdot\g=\d_{x}-\d_y$.\
Similarly we obtain a measure satisfying this property by substituting for $r$ any Lipschitz path from $x$ to $y$.

We will make use of the following construction.

\[lemma:completediv\] Given $S=(x_0,\cdots,x_N)\in {\Omega}^{N+1}$ a sequence of $N+1$ distinct points, there exist a vector measure $\g= U(M_\g,\t_\g,\xi_\g)$ and a finite partition $({\Omega}_i)\subset \A({\Omega})$ of ${\Omega}$ such that

1.  ${\nabla}\cdot\g=-N\d_{x_0}+\sum_{i=1}^{N} \d_{x_i}$,

2.  $\t_\g:M_\g{\rightarrow}\{1,N\}$,

3.  each ${\Omega}_i$ is a polyhedron,

4.  $ M_\g\subset \bigcup_i \de{\Omega}_i$,

5.  ${\Omega}_i$ is of finite perimeter for each $i$ and ${\Omega}_i\cap{\Omega}_j=\emptyset$ for $i\neq j$,

6.  $\L^2({\Omega}\sm\cup_i{\Omega}_i)=0$.

Moreover if $M$ is a $1$ dimensional countably rectifiable set, we can choose $\gamma$ and $({\Omega}_i)$ such that $\H^1(M\cap \bigcup_i \de{\Omega}_i)=0$.

Let us fix a point $p\in{\Omega}\setminus S$. By Example \[ex:div2punti\] we can construct a measure $\g_i$ with ${\nabla}\cdot\g_i=\d_{x_i}-\d_{p}$ for $i\in\{0,\cdots,N\}$. We define $$\g=-N \g_0 +\sum_{i=1}^N\g_i.$$ By 