While studying Hausdorff contents I came across this article by Roy O. Davies. He gives a really nice example in $\mathbb{R}^{2}$ which shows that the 'continuity from below'property (namely,$\mathscr{H}_{2}^{1}(E_{n}) \to \mathscr{H}_{2}^{1}(\cup_{n}E_{n})$ for increasing sequences of sets $\{E_{n}\}$) may or may not hold for $\mathscr{H}_{2}^{1}$ according to the requirements imposed upon the coverings in its definition. Such requirements might differ in the following sense:
In order to construct an increasing sequence of sets, he takes $E_{n}$ to be the intersection between the closed balls $x^{2}+y^{2}\leq 1$ and $(x-2n^{-1})^{2}+y^{2}\leq 1$. Denoting $E=\cup_{n}E_{n}$, which turns out to be $B(0;1)$ together with the right side of its boundary, he claims that $\mathscr{H}_{2}^{1}(E)=4$ under the first three definitions above.
I have two questions:
- Regarding definitions $(i)$ and $(iii)$, how does one get the estimate $\mathscr{H}_{2}^{1}(E)\geq 4$?
- Does the same hold for $\mathscr{H}_{\infty}^{1}(E)$?
Any help would be appreciated.