  A metric measure space $(X,d,\mu)$ is said to satisfy the strong annular
decay condition if there is a constant $C>0$ such that $$
\mu\big(B(x,R)\setminus B(x,r)\big)\leq C\, \frac{R-r}{R}\, \mu (B(x,R)) $$ for
each $x\in X$ and all $0<r \leq R$. If $d_{\infty}$ is the distance induced by
the $\infty$-norm in $\mathbb{R}^N$, we construct examples of singular measures
$\mu$ on $\mathbb{R}^N$ such that $(\mathbb{R}^N, d_{\infty},\mu)$ satisfies
the strong annular decay condition.
