From: edgar@math.ohio-state.edu (G. A. Edgar)
Newsgroups: sci.math
Subject: Re: centroid of the Mandelbrot's set
Date: Wed, 19 Jun 1996 10:31:16 -0500

> 
> Is the Mandelbrot set measurable? I mean, does it *have* an area?
> If so, what is it?

Yes, it is a closed set, so it is measurable.  It has an area (in
the sense of 2-dimensional Lebesgue measure).  The area has been
estimated, but not computed exactly.

> It is smaller than the circle r=2 and bigger than r=1 (or whatever),
> the boundary is a line (although VERY crumpled). I would expect
> it even has a Riemann integral.

In fact this is unknown.  A closed set has an area computed as
a Riemann integral if and only if its boundary has measure zero.
The boundary of the Mandelbrot set has Hausdorff dimension 2, but
it is unknown whether that boundary has postive area.
-- 
Gerald A. Edgar                   edgar@math.ohio-state.edu