David C. Ullrich science forum Guru
Joined: 28 Apr 2005
Posts: 2250

Posted: Tue Jun 27, 2006 3:22 pm Post subject:
Hausdorff Measure



K will always be a compact subset of R^d,
and omega will always be a function from
(0,infinity) to (0, infinity) such that
omega(2r) <= c omega(r). Say m_omega is
the associated Hausdorff measure, and
as usual if a >= 0 let m_a = m_omega
where omega(r) = r^a.
Consider two properties:
(i) m_a restricted to K is sigmafinite.
(ii) m_omega(K) = 0 for all omega such that
omega(r)/r^a > 0 as r > 0.
It's clear that (i) implies (ii). What
about the converse?
I tend to doubt it. Although it's true, at
least for d = 1 and I think for d > 1, when
a = 0 : if K is uncountable then K contains
a topological Cantor set, hence K supports
a continuous probability measure mu. If
omega(r) = sup_{I <= r} mu(r) then
omega(r) > 0 and m_omega(K) > 0. But it
seems likely that that's a very special
case.
************************
David C. Ullrich 
