Outer measure does not satisfy countable additivity

Question

Let $\Omega \subseteq \Bbb R^n$, we define the outer measure the following way : $m^*(\Omega):= \inf\{ \sum_{j \in J} \text{vol } B_j \lvert \{B_j\} \text{ is countable covering by boxes of } \Omega \}$ where a box $B$ is $B:=\Pi_{i=1}^n (a_i,b_i)$ and vol($B)=\Pi_{i=1}^n (b_i-a_i)$.

It is said in my lectures notes that this outer measure does not satisfy countable additivity : if $A,B$ disjoint then $m^*(A \cup B) = m^*(A) + m^*(B)$.

Can you give me a counterexample for this i.e. $A,B$ disjoint sets such that $m^*(A \cup B) \neq m^*(A) + m^*(B)$.

Answer 1

You need to look for weird sets, like a Vitali set.