Uncentered maximal weak-type inequality ($|\{M_u\mu >\alpha \}|\le 3^d\mu ({\mathbb{R}}^d)/\alpha$)

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.