More in dimensions

In my previous post, I briefly discussed the concept of dimensions and the different considerations for topological and fractal dimensions. There is much more to cover on this topic, but my goal is to provide a general understanding of the concept rather than go into the details. $$\\$$ It is from its name "packing", we want to pack disjoint sets in our large set to measure it, but here,we are not talking about covering the set but about "packing".

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Defintion

First, let $E \subset R^{n}$, we define a packing in the sense of Euclidean spaces by the following countable set: $$\\$$ \[ \Pi = \{B_{i}, i \in \mathbf{N} : B_{i}=B(x,r) \textit{ for some }x \in E , r < \delta, B_{i} \cap B_{j} = \emptyset \textit{ such that }i \ne j\}\] $$\\$$ I think it is obvious that this set is not well defined, as we can have many sets satisfy the same definition. Therefore, we will take the set of all sets satisfy these conditions. Call this set $\Psi$. $$\\$$ We define the packing pre-measure by the following: \[P_{\delta} ^{s} (E) = \sup \sum_{\{B(x,r)\} \in \Psi } \alpha(s) r^{s}\] Taking the limit : \[P^{s}(E) = \lim_{\delta \to \infty} P_{\delta} ^{s} (E) = \inf_{\delta>0}P_{\delta} ^{s} (E)\] $P^{s}$ is called the $\textbf{packing pre-measure}$ $$\\$$

But why is it not just a measure?

$$\\$$ That is a good question, and the idea of taking the $\textbf{supremum}$ of the sum above is what makes the case as we see, but we can see also that the idea of packing is only efficient with this concept. $$\\$$ We need to define a measure from this $\textbf{pre-measure}$. $$\\$$ For $E \subset R^{n}$, we define the $\textbf{packing measure}$ as following: \[\bar{P}^{s}(E) = \inf \left\lbrace \sum_{i=1}^{\infty}P^{s}(U_{i}) : E \subset \bigcup_{i=1}^{\infty} U{i} \right\rbrace \] $\textbf{Packing dimensions}$ is defined in the same way the Hausdorff dimensions is defined \[\dim_{P}(E) = \sup\{s:\bar{P}^{s}(E)=\infty\}\]

To be continued.