Example of strictly subadditive lebesgue outer measure

Solution 1:

Take a Vitali set in the unit interval $[0,1)$, and call it $V$. (Define an equivalence relation on $[0,1)$ by $x \sim y$ iff $x - y \in \mathbb{Q}$, and let $V$ be obtained from the Axiom of Choice by choosing exactly one element from every $\sim$-equivalence class.)

By $Q$ I will denote $\mathbb{Q} \cap [0,1)$. For each $q \in Q$ let $V_q$ denote the modulo $1$ shift of $V$ by $q$: $$V_q = \{ x + q : x \in V, x < 1-q \} \cup \{ x + q - 1 : x \in V , 1-q \leq x \}.$$ Note the following facts:

• $V_q \cap V_p = \emptyset$ if $p \neq q$.
• $\bigcup_{q \in Q} V_q = [0,1)$.

(Both of the above follow from the fact that $V$ contains exactly one element of every $\sim$-equivalence class.)

By the second fact above we have that $$\mu^* \left( \textstyle{\bigcup_{q \in Q}} V_q \right) = 1.$$ On the other hand, by the translation invariance of $\mu^*$ is follows that $\mu^* ( V_q ) = \mu^* ( V )$ for all $q$, and since $V$ is not Lebesgue measurable it follows that $\mu^* ( V ) > 0$, and therefore $$\sum_{q \in Q} \mu^* ( V_k ) = + \infty.$$

Solution 2:

A Bernstein set, $A\subseteq[0,1]$ has the property that both $A$ and $[0,1]\setminus A$ have outer measure $1$.