这是一份GLA格拉斯哥大学MATHS5065_1 5/MATHS4112_1作业代写的成功案例
Let $\mu$ be a mass distribution on $F$ and suppose that for some s there are numbers $c>0$ and $\varepsilon>0$ such that
$$
\mu(U) \leqslant c|U|^{s}
$$
for all sets $U$ with $|U| \leqslant \varepsilon$. Then $\mathcal{H}^{s}(F) \geqslant \mu(F) / c$ and
$$
s \leqslant \operatorname{dim}{\mathrm{H}} F \leqslant \operatorname{dim}{\mathrm{B}} F \leqslant \overline{\operatorname{dim}}_{\mathrm{B}} F .
$$
If $\left{U_{i}\right}$ is any cover of $F$ then
$$
0<\mu(F) \leqslant \mu\left(\bigcup_{i} U_{i}\right) \leqslant \sum_{i} \mu\left(U_{i}\right) \leqslant c \sum_{i}\left|U_{i}\right|^{s}
$$
using properties of a measure and (4.1).
Taking infima, $\mathcal{H}{8}^{s}(F) \geqslant \mu(F) / c$ if $\delta$ is small enough, so $\mathcal{H}^{s}(F) \geqslant \mu(F) / c$. Since $\mu(F)>0$ we get $\operatorname{dim}{H} F \geqslant s$.
Notice that the conclusion $\mathcal{H}^{s}(F) \geqslant \mu(F) / c$ remains true if $\mu$ is a mass distribution on $\mathbb{R}^{n}$ and $F$ is any subset.
MATHS5065_1 5/MATHS4112_1 COURSE NOTES ：
Each $k$ th level interval supports mass $\left(m_{1} \cdots m_{k}\right)^{-1}$ so that
$$$$
for every $0 \leqslant s \leqslant 1$.
Hence
$$
\frac{\mu(U)}{|U|^{s}} \leqslant \frac{2^{s}}{\left(m_{1} \cdots m_{k-1}\right) m_{k}^{s} \varepsilon_{k}^{s}}
$$
If
$$
s<\lim {k \rightarrow \infty} \log \left(m{1} \cdots m_{k-1}\right) /-\log \left(m_{k} \varepsilon_{k}\right)
$$