  We prove that the Brouwer degree $\mathrm{deg}(u,U,\cdot)$ for a function
$u\in C^{0,\alpha}( U;\mathbb{R}^n)$ is in $L^p(\mathbb{R}^n)$ if $1\leq
p<\frac{n\alpha}d$, where $U\subset \mathbb{R}^n$ is open and bounded and $d$
is the box dimension of $\partial U$. This is supplemented by a theorem showing
that $u_j\to u$ in $C^{0,\alpha}(U;\mathbb{R}^n)$ implies
$\mathrm{deg}(u_j,U,\cdot)\to \mathrm{deg}(u,U,\cdot)$ in $L^p(\mathbb{R}^n)$
for the parameter regime $1\leq p<\frac{n\alpha}d$, while there exist
convergent sequences $u_j\to u$ in $C^{0,\alpha}(U;\mathbb{R}^n)$ such that
$\|\mathrm{deg}(u_j,U,\cdot)\|_{L^p}\to \infty$ for the opposite regime
$p>\frac{n\alpha}d$.
