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