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 ${\bf P}$ be the set of all ordered tuplets of distinct elements of $[a,b]$ whose entries are increasing:

{\bf P}=\{(t_{1},\ldots t_{n})\mid(a\leq t_{1}<t_{2}<\cdots<t_{n}\leq b)\wedge% (n\in\mathbb{Z})\wedge(n>0)\}

We shall refer to elements of ${\bf P}$ as partitions of the interval $[a,b]$ . We shall say that $(t_{1},\ldots,t_{n})$ is a refinement of a partition $(s_{1},\ldots,s_{m})$ if $\{t_{1},\ldots,t_{n}\}\supset\{s_{1},\ldots,s_{m}\}$ . Let ${\bf F}\subset\mathcal{P}({\bf P})$ be the set of all subsets of ${\bf P}$ such that, if a certain partition belongs to ${\bf F}$ then so do all refinements of that partition.

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

Next, note that, if a partition of $B$ is a refinement of a partition of $A$ then, by the triangle inequality, the length of $\Pi(B)$ is greater than the length of $\Pi(A)$ . By definition, for every $\epsilon>0$ , we can pick a partition $A$ such that the length of $\Pi(A)$ differs from the length of the curve by at most $\epsilon$ . Since the length of $\Pi(B)$ for any partition $B$ refining $A$ lies between the length of $\Pi(A)$ and the length of the curve, we see that the length of $\Pi(B)$ will also differ by at most $\epsilon$ , so the length of the curve is the limit of the length of polygonal lines according to the filter generated by ${\bf F}$ .

Title	arclength as filtered limit
Canonical name	ArclengthAsFilteredLimit
Date of creation	2013-03-22 15:49:34
Last modified on	2013-03-22 15:49:34
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	13
Author	rspuzio (6075)
Entry type	Result
Classification	msc 51N05