TITLE: Do Integrals over Fractals Exist?
QUESTION [32 upvotes]: Given, for example, a line integral like 
$$
\int_\gamma f \; ds
$$
with $f$ not further defined, yet.

What happens, if the contour $\gamma$ happens to be a fractal curve?

Since all fractal curves (to my knowledge) are not differentiable and have an infinite length, we should run into some trouble. Further questions of interest are:

What happens if the integral is taken over a Julia Set?
What happens in the case of a space-filling curve? Do we get a surface integral then?

I'd be glad, if anybody could wrap her/his 2.79-Hausdorff-dimensional brain surface around it!

REPLY [6 votes]: Sure, there are several ways to integrate over fractals. Jenny Harrison has worked out a lot using chains, Boris Katz has some papers you could look up... there is this using fractal measures http://arxiv.org/pdf/chao-dyn/9804006.pdf, and http://www.minet.uni-jena.de/geometrie/publicat/pdffiles/iwrtff1_ptrf98.pdf using fractional calculus. I have been formulating methods using the hyperreal number system, where, for instance, the limit case of the Cantor middle thirds set is a set of closed infinitesimal length intervals. An integral over infinitesimal elements can be defined, then the integrals summed over the whole fractal. See also "Analysis on Fractals" by Robert Strichartz.
John Starrett