Length of metric spaces and rectifiable mappings
by
Jan Hejcman
University of Education, Hradec Králové, Czech Republic

Definition Let (X, , d) be a metric space. If A subset X, A is finite, we put lng A = 0 for A = \emtyset and, if A =/= \emptyset we put lng A = min\sum_i=1ⁿ d(x_i-1, x_i) where the minimum is taken over all finite sequences (x_i)_i=0ⁿ such that {x_i;i=0, ..., n} = A. Finally, for any Y subset X, we put

lng Y = sup {lng A ; A subset Y, A is finite }.

We will say that lng Y is the length of Y.

D. H. Fremlin published a series of papers concerning spaces of finite length. The length in his sense is the one-dimensional Hausdorff outer measure. His finite length is a weaker condition. Rectifiable mapping is a natural generalization of function of bounded variation. Our concept of length seems to be useful because, e.g.: A metric space X is rectifiable image of a subset of R if and only if lng X is finite. It is easy to show that if lng X, lng Y are finite then lng[`X], lng (X \cup Y) are finite, X is totally bounded, but lng (X×Y) may be inifinite. The following problems will be discussed:
Alternative definition of length by means of the length of scelets of some graphs.
Spaces with powers of finite length.
Which spaces (of finite length) are rectifiable images of some special subsets of R.

Date received: June 3, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabc-23.