  We show that the uncentered Hardy-Littlewood maximal operators associated
with the Radon measure $\mu$ on $\mathbb{R}^d$ have the uniform lower
$L^p$-bounds (independent of $\mu$) that are strictly greater than $1$, if
$\mu$ satisfies a mild continuity assumption and $\mu(\mathbb{R}^d)=\infty$. We
actually do that in the more general context of metric measure space
$(X,d,\mu)$ satisfying the Besicovitch covering property. In addition, we also
illustrate that the continuity condition can not be ignored by constructing
counterexamples.
