Is finite additivity a property of the Lebesegue outer measure? In the Lebesgue outer measure we have the following properties: the length of the empty set is $0$, monotonicity and countable subadditivity.
My question is, is it possible to use the above properties to show finite additivity?
I have no idea where to begin (or even if it's true) so any hints are definitely appreciated.
 A: Lebesgue outer measure is in fact not finitely additive. This can be shown via the usual construction of a Vitali decomposition of $[0, 1)$ - we can partition $[0, 1)$ into countably many sets $V_i$ which have the same outer measure $\kappa$. By countable subadditivity (and the fact that $[0, 1)$ doesn't have measure zero), we have $\kappa>0$; but then $m^*([0, 1))=1<\sum_{i\in\mathbb{N}}m^*(V_i)=\infty$.
(Note that the $V_i$s are disjoint - that's implied by the word "partition".)

How do we get a Vitali decomposition in the first place? If you haven't seen this before, it's a neat argument! Consider the equivalence relation $\sim$ on $[0, 1)$ given by $$a\sim b\iff a-b\in\mathbb{Q}.$$ Let $V\subseteq[0, 1)$ be a transversal of this equivalence relation - that is, $V$ contains exactly one element of each equivalence class. 
Note that in order to get $V$ we need the axiom of choice. Indeed, without AC it is consistent that every set is measurable!
Now, if you imagine wrapping $[0, 1)$ into a circle, the rational translates of $V$ partition $[0, 1)$! Formally, for $q\in\mathbb{Q}\cap [0, 1)$ let $V+q=\{v+q: v\in V\}$, and let $$V_q=\{v+q: v+q\in [0, 1)\}\cup\{v+q-1: v+q\in [1, 2)\}$$ be the "wrapped around" version of $V+q$. Then you can show that:


*

*The $V_q$s partition $[0, 1)$,

*$m^*(V_p)=m^*(V_q)$ for $p, q\in\mathbb{Q}\cap [0, 1)$ (this is basically translation-invariance of $m^*$), and

*$m^*(V_p)>0$ for every $p\in\mathbb{Q}\cap [0, 1)$ (this uses the facts that $\mathbb{Q}$ is countable, the countable union of measure-zero sets is measure-zero, and $[0, 1)$ is not measure zero).
This is, I believe, the historically first example of a non-measurable set. (There are others.)
