Countable Subadditivity of outer measure. Because of the countable subaadditivity of outer measure, we have
$m^*(\cup^{\infty}_{k=1}) \leq \sum^{\infty}_{k=1}m^*(E_k)$. I'm just wondering, when does the equality happen? Do we only get the equality if it is Lebesgue measurable? In other words, if $\cup^{\infty}_{k=1} E_k$ is not measurable, do we always have a strict inequality? 
 A: If you think about nonmeasurable sets that are "sufficiently disjoint", you can still have equality. For example, let $A_1$ be a nonmeasurable set in the interval $[2,3],$ let $A_2$ be a nonmeasurable set in the interval $[4,5],$ ..., let $A_n$ be a nonmeasurable set in the interval $[2n,\; 2n+1],$ ... Then you can show that the outer measure of the union of these sets is the sum of the outer measures of these sets. One way of proving this makes use of: (1) Sufficently "refined" open interval covers of the union can be split into disjoint open interval covers of the individual sets. (2) The union of any collection of open interval covers, each of which is an open interval cover of one of the sets (and all of the sets get used up in this way), gives rise to an open interval cover of the union.
More generally, if the sets are pairwise metrically separated, then the outer measue of the union of the sets will be the sum of the outer measures of the sets, where $A$ and $B$ are said to be metrically separated if
$$\inf\{|x-y|:\; x \in A,\;\; y \in B\}\; > \; 0.$$
ADDED 2 DAYS LATER I wrote the above without access to any of my books or notes. Yesterday, while in bed (trying to sleep . . .), I wondered whether we need to have a positive lower bound on all the "metric separation" infimum constants. Later, I looked in some of my books and notes. After doing this I decided that I should add some additional comments.
1. At the present time a much more common term for "metrically separated" is positively separated. Moreover, one says that an outer measure is a metric outer measure if it is countably additive on positively separated sets. Lebesgue outer measure is an example of a metric outer measure.
2. To prove that Lebesgue outer measure is countably additive on positively separated sets, it is enough to prove it is finitely additive on positively separated sets (do it for 2 sets, then use mathematical induction) and then apply continuity from below to get countable additivity on positively separated sets. In particular, we do not need the existence of a positive lower bound on all the "positive set separation" infimum constants. Here's the basic idea behind one way to show that Lebesgue outer measure is additive for 2 positively separated sets. First, note that for each $\delta > 0,$ each open ball in ${\mathbb R}^n$ can be written as the union of countably many open balls each of which has diameter less than $\delta.$ Therefore, given any countable covering of $A \cup B$ by open balls, we can replace this with a countable covering of $A \cup B$ by open balls each of which has diameter less than $\inf\{|x-y|:\; x \in A,\;\; y \in B\}.$ Note that none of these replacement open balls can intersect more than one of the sets $A$ and $B.$
