(see Theorem 1.59 in [@AFP]), that, possibly after extracting a further subsequence $r_{k}$, $$\mu_{k}\rightharpoondown\mu_{\infty}\quad\mbox{in}\quad\mathcal{M}\,,$$ Altogether we have for the sequence $r_{k}$ $$u_{k}\rightarrow u_{\infty}\quad\mbox{in}\quad L^{1}_{loc}(\mathbb{R}^2)\,$$ and $$\mu_{k}\rightharpoondown\mu_{\infty}\quad\mbox{in}\quad\mathcal{M}_{loc}(\mathbb{R}^{2})\,,$$ which is what we aimed to prove.

We argue by contradiction. Therefore we assume that there exists a point $(x_{0},t_{0})$ such that $$\label{contradiction}
\limsup_{r\rightarrow 0^+}\frac{1}{r}\int_{B_{r}((x_{0},t_{0}))}\int_{\mathbb{R}}m(x,t,a)\,da\,dx\,dt<0\,.$$ For a sequence $r_{n}\rightarrow0^{+}$ we define $$\begin{array}{c}
u_{n}(x,t):=\left(D_{n}^{-1}\right)^{*}u(x,t)\\[5mm]
\mbox{and}\\[5mm]
\mu_{n}:=\displaystyle\frac{1}{r_n}\int_{\mathbb{R}}\left(D_n\right)_*m\,da\,.
\end{array}$$ Let $u_{k}$ and $\mu_{k}$ be the subsequences given by Lemma \[compactness\], with limits $u_{\infty}$, $\mu_{\infty}$. Then we have by strong convergence, that $u_{\infty}$ is a weak solution of $$\partial_{t}u_{\infty}+\partial_{x}f(u_{\infty})=0\,.$$ Furthermore, by the uniqueness of the distributional limit, we conclude that $$\mu_{\infty}=\int_{\mathbb{R}}\partial_{t}(u_{\infty}\wedge a)+\partial_{x}f(u_{\infty}\wedge a)\,da\,.$$ From (\[contradiction\]) we want to conclude now, that $$\label{negative_of_limit}
\mu_{\infty}(B_R(0,0))<0\quad\mbox{for all}\quad R>0\,.$$

**Proof of (\[negative\_of\_limit\]).** For the sake of contradiction, we assume, that there exists a $R_0$ such that $$\mu_{\infty}(B_{R_0}(0,0))\geq0\,.$$ In [@Le] it is proved, that there exits a set $K$, which is either a line, or a half-line, or the empty set, such that $$\label{structure_of_measure}
 \partial_{t}u_{\infty}\wedge a+\partial_{x}f(u_{\infty}\wedge a)=\left((X(u_{\infty}^{+}\wedge a)-X(u_{\infty}^{-}\wedge a)\right)\cdot\omega_{K}\,\mathcal{H}^{1}\res K\,,$$ where $$\label{vectorfield_X}
 X(u)=\begin{pmatrix}f(u)\\ u\end{pmatrix}\quad\mbox{and}\quad  \omega_{K}=\frac{|u_{\infty}^{+}-u_{\infty}^{-}|}{|X(u^{+}_{\infty})-X(u_{\infty}^{-})|}\begin{pmatrix}1\\ -\frac{f(u_{\infty}^{+})-f(u_{\infty}^{-})}{u_{\infty}^{+}-u_{\infty}^{-}}\end{pmatrix}\,.$$ Moreover therein it is proved, that $u_{\infty}$ is $\mathcal{H}^{1}$-a.e. approximately continuous in $K^{c}$ and has $\mathcal{H}^{1}$-a.e. constant approximate jump points $u_{\infty}^{\pm}$ on $K\,$.

A short calculation reveals 