Definition 24.7.4 (Hardy-Littlewood Maximal Function).label Let $\mu: \cb_{\real^d}\to [0, \infty]$ be a regular Borel measure on $\real^{d}$, then
\[H\mu: \real^{d} \to [0, \infty) \quad x \mapsto \sup_{r > 0}A_{r}\mu(x)\]
is the Hardy-Littlewood maximal function of $\mu$.
If $\mu(\partial B_{\real^d}(x, r)) = 0$ for all $x \in \real^{d}$ and $r > 0$, then $H\mu$ is lower semicontinuous, and in particular Borel measurable.
Proof. If $\mu(\partial B_{\real^d}(x, r)) = 0$ for all $x \in \real^{d}$ and $r > 0$, then $(r, x) \mapsto A_{r}\mu(x)$ is jointly continuous by Lemma 24.7.3. Since $H\mu$ is a supremum of continuous functions, it is lower semicontinuous by Proposition 5.22.3.$\square$
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