arclength as filtered limit

The length ( of a rectifiable curve may be phrased as a filtered limit. To do this, we will define a filter of partitions of an intervalMathworldPlanetmathPlanetmath [a,b]. Let 𝐏 be the set of all ordered tuplets of distinct elements 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 (t1,,tn) is a refinementPlanetmathPlanetmath of a partition (s1,,sm) if {t1,,tn}{s1,,sm}. Let 𝐅𝒫(𝐏) be the set of all subsets of 𝐏 such that, if a certain partition belongs 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 AB𝐅.

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