= Uncentered maximal weak-type inequality
{title2=$|\{M_u\mu>\alpha\}|\leq3^d\mu(\mathbb R^d)/\alpha$}
For a finite positive measure on Euclidean space, cover a compact subset of the maximal superlevel set by finitely many high-density balls. <Wiener covering lemma> selects disjoint balls whose triples cover it. Their total measure mass is at most the mass of the original measure. <Inner regularity of Lebesgue measure> yields the displayed bound. Applied to approximation errors, it proves the <Lebesgue differentiation theorem>.
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