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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 sigma-finite.

(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

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

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