Petersburg Department of Steklov Institute of Mathematics

PREPRINT 9/1999

MEASURABILITY OF SELF-SIMILAR SPECTRAL GEOMETRIES

This preprint was accepted May, 1999
Contact: S.V.Buyalo

ABSTRACT:
The notion of a spectral geometry on a compact metric space
$X$
is motivated, on the one hand, by tasks of calculus on general metric spaces
and, on the other hand, by the notion of a spectral triple playing the role of
a ``Riemannian manifold'' in noncommutative geometry. A spectral geometry
$M$
is given by a symmetric subset
$B\subset X^2\setminus\De$
which is finite outside of each neighborhood of the diagonal
$\De$,
and defines, via the Dixmier trace
$\dtr$,
a Radon measure
$d_{\om}M$
on
$X$
converting
$X$
into a metric measure space. The geometry
$M$
is called
$\om$-measurable,
if the measure
$d_{\om}M$
is finite and independent of the choice of the limiting procedure
$\om$.
We prove the
$\om$-measurability
of a broad class of self-similar geometries including geometries on each
self-similar compact subset in
$R^n$
satisfying the standard OSC (Open Set Condition).