Lower bounds for uncentered maximal functions on metric measure space

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 $\mu ({\mathbb{R}}^d)=\mathrm{\infty }$. 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.