In the book "Harmonic Measure" by Garnett and Marshall, we have the following result:


**Lemma I.2.3** Let $\mu$ be a positive Borel measure on $\partial{\mathbb{D}}$ and let $\{I\_{j}\}$ be a finite sequence of open intervals in $\partial{\mathbb{D}}$. Then $\{I\_{j}\}$ contains a pairwise disjoint subfamily $\{J\_{k}\}$ such that


$\sum\limits\_{k}\mu(J\_{k})\geq{\frac{1}{3}\mu\big(\cup\_{j}I\_{j}\big)}$


By repeating the argument in Garnett and Marshall we get the same result for positive measures $\mu$ on $\mathbb{R}$, only with a better constant- $1/2$ instead of $1/3$. Does an analogous result hold in $\mathbb{R}^{n}$ for $n>1$? Namely, does there exist an absolute constant $c=c(n)>0$ such that if $\mu$ is a positive Borel measure on $\mathbb{R}^{n}$ and $\{Q\_{i}\}$ is a finite sequence of balls or cubes, then there exists a pairwise disjoint subfamily of balls or cubes $\{J\_{k}\}$ such that


$\sum\limits\_{k}\mu(J\_{k})\geq{c\cdot{}\mu\big(\cup\_{i}Q\_{i}\big)}$


If this fails in higher dimensions, does there exist a characterization of the measures for which such a condition holds?



A Covering Lemma for Arbitrary Measures
The answer is no for $n \ge 2$. Consider open cubes $Q(c,1)=\{c\_i <x\_i <c\_i+1\}$ of side $1$. Starting from $Q(0,1)$ and moving $c$ along the diagonal joining $0$ with $(1,\dotsc,1)$ one constructs $N$ cubes $Q(c\_i,1)$ such that two of them always intersect and no cube is contained in the union of the others. Take $$z\_i \in Q(c\_i,1) \setminus \bigcup\_{j \neq i}Q(c\_j,1)$$ and $\mu$ equal to the sum of Dirac masses at $z\_i$. Then $\mu(\bigcup Q(c\_i,1))=N$ but any disjoint subcollection of the cubes consist of only $1$ cube and has measure $1$.


