Vitali set proof: Why is $1 \leq \sum_{k \in \mathbb{N}}m(\mathcal{N}_k) \leq 3$ impossible?
Solution 1:
Since $m(\mathcal N_k)=m(\mathcal N)$ for each $k$, both cases lead to a contradiction:
- If $m(\mathcal N)=0$, then the infinite sum equals $0$ as well, yielding $1\leq 0\leq 3$, which is absurd.
- If $m(\mathcal N)>0$, the infinite sum equals infinity, which gives $1\leq\infty\leq 3$, again a contradiction.