On Neugebauer's covering theorem (Q1651478)

The author revisits a covering result of \textit{C. J. Neugebauer} [Proc. Amer. Math. Soc. 130, No. 10, 2883--2891 (2002; Zbl 1003.42012)]. The original result of Neugebauer concerns finite measures \(\mu\) in \(\mathbb R^n\) which are absolutely continuous with respect to the Lebesgue measure. It states that for every \(\epsilon>0\) and for every collection of closed cubes \(\{Q_\alpha\}\) with \(\mu(\bigcup_\alpha Q_\alpha)>\epsilon\) there exists a finite disjoint subcollection \(\{\tilde Q_\beta\}\subseteq \{Q_\alpha\}\) with \(\mu(\bigcup_\beta \tilde Q_\beta)\leq C \mu(\bigcup_\alpha Q_\alpha)\). The constant \(C\) depends on \(\epsilon\) and the density of \(\mu\). In the paper under review the author generalizes this result in appropriate geometrically doubling metric spaces \((X,d)\). Additionally the author assumes that for every pair of points in the space there is an approximate midpoint. Further let \(\mu\) is a measure on \((X,d)\) such that for all balls \(B(x,r)\) we have that \[ ch(r)\leq m(B(x,r))\leq h(r) \qquad\forall r>0, \] where \(h:(0,\infty)\to (0,\infty)\); the constant \(c\) and function \(h\) are independent of \(x\in X\) and \(r>0\). Then an appropriate generalization of Neugebauer's covering result holds true. The argument in the current paper uses the Lebesgue differentiation theorem while the original argument of Neugebauer used the Vitali covering lemma. The proof of the result gives some explicit dependence of the constant \(C\) on \(f\) and \(\epsilon>0\). As an application the author provides an upper bound (of weak \((1,1)\) numerology) for the level sets of the uncentered maximal function defined with respect to a measure \(f(x)d\mu(x)\) on \((X,d)\) (where \(\mu\) and \((X,d)\) are as above) applied to sets of \(\nu\)-measure bounded away from \(0\).