The Vitali covering lemma states that given a vitali cover, V, of closed balls of a set A, then given $\displaystyle \epsilon > 0 $ there exists a disjoint collection of balls {$\displaystyle V_k $} such that:

1. $\displaystyle m(\coprod V_k) \leq m^*(A) + \epsilon $ Note: I'm using $\displaystyle \coprod $ for disjoint union, I'm not sure if that's the correct symbol.
2. $\displaystyle m(A\setminus\coprod V_k) = 0 $

I know how to prove it if A is bounded. I am now trying to prove it if A is unbounded. What I have so far is that we let A = $\displaystyle \coprod A_i $ where each $\displaystyle A_i $ is bounded. Then, for each i we can find a set of disjoint balls in V, {$\displaystyle V^i_k $} such that:

1. $\displaystyle m(\coprod_k V^i_k) \leq m^*(A_i) + \frac{\epsilon}{2^i} $
2. $\displaystyle m(A_i\setminus\coprod_k V^i_k) = 0 $

The problem now is that if I take the union of all the balls, then the equations work, but I'm pretty sure its possible that all the balls are not disjoint, and I'm not sure how to fix this.