这是一份manchester曼切斯特大学 MATH31042作业代写的成功案例
Fundamental to most definitions of dimension is the idea of ‘measurement at scale $\delta$ ‘. For each $\delta$, we measure a set in a way that ignores irregularities of size less than $\delta$, and we see how these measurements behave as $\delta \rightarrow 0$. For example, if $F$ is a plane curve, then our measurement, $M_{\delta}(F)$, might be the number of steps required by a pair of dividers set at length $\delta$ to traverse $F$. A dimension of $F$ is then determined by the power law (if any) obeyed by $M_{\delta}(F)$ as $\delta \rightarrow 0$. If
$$
M_{\delta}(F) \sim c \delta^{-s}
$$
for constants $c$ and $s$, we might say that $F$ has ‘divider dimension’ $s$, with $c$ regarded as the ‘s-dimensional length’ of $F$. Taking logarithms
$$
\log M_{\delta}(F) \simeq \log c-s \log \delta
$$
in the sense that the difference of the two sides tends to 0 with $\delta$, and
$$
s=\lim {\delta \rightarrow 0} \frac{\log M{\delta}(F)}{-\log \delta}
$$
MATH31042 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 $F$.