🚀 go-pugleaf

Newsgroups: sci.math,
Subject: sci.math.research, sci.fractals
Summary: 
Followup-To: 
Distribution: usa
Organization: Rensselaer Polytechnic Institute, Troy NY
Keywords: Lebesgue measure, Cantor set, Fractals, Hausdorff dimension


   In Real Analysis, I learned that the set {x : 0 < x < 1} has Lebesgue 
measure 1 for the Lebesgue measure defined in R^1, but is 0 for the Lebesgue
measure defined in R^2 (thus, the measure of a set is relative to what 
space, i.e., R^1, R^2, etc..., the Lebesgue measure is defined in).

   Now, the Cantor set has Lebesgue measure 0 for the Lebesgue measure defined
in spaces R^1, R^2, etc...  However, the Cantor does have a fractal dimension
(or Hausdorff dimension) between 0 and 1 (I cannnot recall the value), and
I'll call this dimension Df.  Is it possible to define a Lebesgue measure in
a space of dimension Df (R^Df) such that the measure of the Cantor set is a 
value other than 0?

                        John