arclength as filtered limit

The length (http://planetmath.org/Rectifiable) of a rectifiable curve may be phrased as a filtered limit.To do this, we will define a filter of partitions of an interval$[a,b]$. Let $𝐏$ be the set of all ordered tuplets of distinctelements of $[a,b]$ whose entries are increasing:

We shall refer to elements of $𝐏$ as partitions of the interval$[a,b]$. We shall say that $(t_1,\mathrm{\dots },t_n)$ is a refinement of apartition $(s_1,\mathrm{\dots },s_m)$ if $\{t_1,\mathrm{\dots },t_n\}\supset \{s_1,\mathrm{\dots },s_m\}$. Let $𝐅\subset 𝒫(𝐏)$ be theset of all subsets of $𝐏$ such that, if a certain partitionbelongs to $𝐅$ then so do all refinements of that partition.

Let us see that $𝐅$ is a filter basis. Suppose that $A$ and $B$are elements of $𝐅$. If a partition belongs to both $A$ and $B$then every one of its refinements will also belong to both $A$ and$B$, hence $A\cap B\in 𝐅$.

Next, note that, if a partition of $B$ is a refinement of a partitionof $A$ then, by the triangle inequality, the length of $\mathrm{\Pi }(B)$ isgreater than the length of $\mathrm{\Pi }(A)$. By definition, for every$ϵ>0$, we can pick a partition $A$ such that the length of$\mathrm{\Pi }(A)$ differs from the length of the curve by at most $ϵ$.Since the length of $\mathrm{\Pi }(B)$ for any partition $B$ refining $A$ liesbetween the length of $\mathrm{\Pi }(A)$ and the length of the curve, we seethat the length of $\mathrm{\Pi }(B)$ will also differ by at most $ϵ$, sothe length of the curve is the limit of the length of polygonal linesaccording to the filter generated by $𝐅$.