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- Jun 20, 2014

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Suppose $\mu$ is a finite Borel measure on $\Bbb R^n$. Define the maximal function of $\mu$ by $$\mathcal{M}\mu(x) = \sup_{0 < r < \infty} \frac{\mu(B(x;r))}{m(B(x;r))}\quad (x\in \Bbb R^n)$$ Here, $m$ denotes the Lebesgue measure on $\Bbb R^n$. Show that if $\mu$ is mutually singular with respect to $m$ (i.e., $\mu \perp m$), then $\mathcal{M}\mu = \infty$ a.e. $[\mu]$.

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