Outer Measure Subadditivity

Question

I'm having trouble constructing a sequence $\{E_n\}$ of disjoint subsets of $\mathbb{R}$ such that $$m^{*}\left(\bigcup_{i}E_i\right) < \sum_{i}m^{*}(E_i).$$ What's a way to gain some intuition here? I'm just kind of playing around with sets, without any real understanding of what these sets might look like, besides the fact that they (probably) are pairwise "close." Please don't provide an answer, just a way to think about this.

Answer 1

Given $\varepsilon >0$ we want to choose, for each $i\geq 1$ a sequence of open and bounded intervals $((a_{i_{k}},b_{i_{k}}))_{k\geq 1}$ that covers $E_{i}$ and satisfies $$\sum_{k=1}^{\infty}(b_{i_{k}}-a_{i_{k}}) < m^{*}(E_{i})+\frac{\varepsilon}{2^{i}}$$ This will give you a little push towards the right direction.

Answer 2

@TheOscillator I think this should be right. Denote the standard Vitali set in $[0,1]$ by $V$, where two points $x,y\in V$ are rationally equivalent iff $x-y \in \mathbb{Q}$. We pick an element of each equivalence class by the Axiom of Choice. Then consider $$U = \bigcup_n V_n=\bigcup_n\left\{v+\frac{1}{n}:v \in V\right\}.$$$U \subset [0,2]$, hence $\mu^{*}(U)\le2$. However, for each $n = n_0$, $\mu^*(V_{n_0})>0$ because of non measurability and translation invariance of the outer measure. Thus $$\sum_n \mu^*(V_n) \to \infty,$$as desired.