Length of Metric Spaces and Rectifiable Mappings
by
Jan Hejcman

Definition. Let (X, d) be a metric space. If A subset X A is finite, we put lngA = 0 for A = \emptyset and, if A =/= \emptyset, we put lngA = 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, 1, ..., n}=A. Finally, for any Y subset X, we put

lngY = sup {lngA : A subset Y, A is finite}.

We will say that lngY 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 outer Hausdorff measure. On the other hand, the one-dimensional Hausdorff measure of N is zero while lngN = \infty.

A natural generalization of function of bounded variation is rectifiable mapping. Our concept of length seems to be useful because, e.g.:

A metric space X is a rectifiable image os a subset of the real line if and only if lngX is finite.

It is easy to show that if lngX, lngY are finite, then lng[`X], lng(X \cup Y) are finite, X is totally bounded, but lng(X ×Y) may be infinite. We will exhibit examples of countable spaces X_m for m in N such that lngX_mⁿ < \infty for n <= m and lngX_mⁿ = \infty for n > m.

Further, rectifiable images of various sets will be characterized as some spaces of finite length.

Date received: June 24, 1996

Copyright © 1996 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 Mathematical Conference Abstracts. Document # caah-36.