(\[scl\]), which satisfies (\[minimality\]). Then for $\mathcal{H}^1$ almost every $(x_{0},t_{0})\in\mathbb{R}\times(0,T)$ $$\limsup_{r\rightarrow 0^+}\frac{1}{r}\int_{\mathbb{R}}m\left(B_r(x_0,t_0),a\right)\,da\geq 0\,.$$

A useful lemma that will be used to prove Lemma \[lemma\_positiv\_density\] is the following.

\[compactness\] Let $u\in L^{\infty}(\mathbb{R}\times(0,T))$ be a weak solution of (\[scl\]). Let $r_{n}\rightarrow0^+$. For $(x_{0},t_{0})\in\mathbb{R}\times(0,T)$ define $$\begin{array}{c}
u_{n}(x,t):=\left(D_n^{-1}\right)^{*}u(x,t)\\[5mm]
\mbox{and}\\[5mm]
\mu_{n}:=\displaystyle\frac{1}{r_n}\int_{\mathbb{R}}\left(D_n\right)_*m\,da\,,
\end{array}$$ where $$\label{dilation}
D_n(x,t)=\left(\frac{x-x_0}{r_n},\frac{t-t_0}{r_n}\right)\,.$$ Then there exists for $\mathcal{H}^1$ almost every $(x_{0},t_{0})\in\mathbb{R}\times(0,T)$ a subsequence $r_{k}$ such that $$u_{k}\rightarrow u_{\infty}\quad\mbox{in}\quad L^{1}_{loc}(\mathbb{R}^2)\,.$$ And furthermore $$\mu_{k}\rightharpoondown\mu_{\infty}\quad\mbox{in}\quad\mathcal{M}_{loc}(\mathbb{R}^{2})\,.$$ Which means in other words $$\int_{\mathbb{R}^2}\psi\,d\mu_k\rightarrow\int_{\mathbb{R}^2}\psi\,d\mu_{\infty}\quad\mbox{for all}\quad\psi\in C^{0}_c(\mathbb{R}^2)\,.$$

Lemma \[compactness\] will be a consequence of of the following proposition, which is proved in Appendix A of [@Le].

\[prop\_lecumberry\] For any constant $M\geq0$, for any bounded set $\Omega$, the set $$\left\{u\in L^{\infty}(\Omega):~\|u\|_{\infty}+\int_{\Omega}\int_{\mathbb{R}}|m(x,t,a)|\leq M\right\}$$ is compact in $L^{1}(\Omega)$ with respect to the strong topology.

By construction we already have $$\label{L_infinity_bound_for_u}
 \|u_{n}\|_{\infty}\leq\|u\|_{\infty}\,.$$ For this reason it remains to show that for all $R>0$ $|\mu_{n}|(B_{R}(0,0))$ and for $\mathcal{H}^1$ almost every $(x_{0},t_{0})\in\mathbb{R}\times(0,T)$ there exists a constant $C>0$, such that $$\label{bound_measure}
 \limsup_{n\rightarrow\infty}\mu_{n}(B_{r_n}(0,0))\leq C\,.$$ But this is a direct consequence of Theorem 2.56 in [@AFP]. Since (\[L\_infinity\_bound\_for\_u\]) and (\[bound\_measure\]) hold, the assumptions of Proposition \[prop\_lecumberry\] are fulfilled and we can extract a subsequence $r_{k'}$ such that $$u_{k'}\rightarrow u_{\infty}\quad\mbox{in}\quad L^{1}_{loc}(\mathbb{R}^2)\,.$$ Additionally we have by the weak$^{*}$ compactness of measures 