If $\mu$ is a doubling Radon measure on $\mathbf{R}^n$, it is well known that for any locally integrable function $f$, the values
$$ (A_\delta f)(x) = \frac{1}{\mu(B(x,\delta))} \int_{B(x,\delta)} f(t) d\mu(t) $$
converges pointwise $\mu$ almost everywhere to $f(x)$. Are there any simple examples of non doubling absolutely continuous measures $\mu$ where this phenomenon fails to occur?