Decomposition of $\Bbb R^n$ as union of countable disjoint closed balls and a null set This is a problem in Frank Jones's Lebesgue integration on Euclidean space (p.57), 

$$\mathbb{R}^n = N \cup \bigcup_{k=1}^\infty \overline{B}_k$$
where $\lambda(N)=0$, and the closed balls are disjoint.

could any one give some hints?

This is an idea which I cannot see if it ends up working or not. Maybe someone can?
Consider the set $\mathcal S$ of all families of closed balls in $\mathbb R^n$ which are pairwise disjoint. Ordered by inclusion, this poset satisfies the hypothesis in Zorn's lemma, so there exist maximal elements $S\in\mathcal S$.
One could hope for $S$ to be a candidate...
Now: is $\mathbb R^n\setminus\bigcup_{B\in S}B$ a null set? I don't see how to prove this...
Notice, though, that every decomposition like those gylns wants does give a maximal element in $\mathcal S$.
Later: This idea does not work: Mike has explicitely constructed a counterexample in the comments below.
Is there a way to fix this? I mean: can one select a smaller set $\mathcal S$  such that the union of its maximal elements have null complement?
