这是一份anu澳大利亚国立大学MATH3062/MATH6116的成功案例

$$
N_{\delta / n}(F) \leqslant N_{\delta}^{\prime}(F) .
$$
If $\delta \sqrt{n}<1$ then
$$
\frac{\log N_{\delta / n}(F)}{-\log (\delta \sqrt{n})} \leqslant \frac{\log N_{\delta}^{\prime}(F)}{-\log \sqrt{n}-\log \delta}
$$
so taking limits as $\delta \rightarrow 0$
$$
\operatorname{dim}{\mathrm{B}} F \leqslant \lim {\delta \rightarrow 0} \frac{\log N_{\delta}^{\prime}(F)}{-\log \delta}
$$
and
$$
\overline{\operatorname{dim}}{\mathrm{B}} F \leqslant \varlimsup{\delta \rightarrow 0} \frac{\log N_{\delta}^{\prime}(F)}{-\log \delta} .
$$
On the other hand, any set of diameter at most $\delta$ is contained in $3^{n}$ mesh cubes of side $\delta$ (by choosing a cube containing some point of the set together with its neighbouring cubes). Thus
$$
N_{\delta}^{\prime}(F) \leqslant 3^{n} N_{\delta}(F)
$$

MATH3029/MATH6109 COURSE NOTES ：

(Note that $\mathcal{H}^{s, 0}$ is just a minor variant of $s$-dimensional Hausdorff measure where only rectangles are allowed in the $\delta$-covers.) The dimension print, print $F$, of $F$ is defined to be the set of non-negative pairs $(s, t)$ for which $\mathcal{H}^{s, t}(F)>0$.
Using standard properties of measures, it is easy to see that we have monotonicity
$$
\text { print } F_{1} \subset \text { print } F_{2} \text { if } F_{1} \subset F_{2}
$$
and countable stability
$$
\operatorname{print}\left(\bigcup_{i=1}^{\infty} F_{i}\right)=\bigcup_{i=1}^{\infty} \text { print } F_{i} \text {. }
$$
Moreover, if $(s, t)$ is a point in print $F$ and $\left(s^{\prime}, t^{\prime}\right)$ satisfies
$$
\begin{gathered}
s^{\prime}+t^{\prime} \leqslant s+t \
t^{\prime} \leqslant t
\end{gathered}
$$
then $\left(s^{\prime}, t^{\prime}\right)$ is also in print $F$.