  Given a spectral triple (A,D,H), the functionals on A of the form a ->
tau_omega(a|D|^(-t)) are studied, where tau_omega is a singular trace, and
omega is a generalised limit. When tau_omega is the Dixmier trace, the unique
exponent d giving rise possibly to a non-trivial functional is called Hausdorff
dimension, and the corresponding functional the (d-dimensional) Hausdorff
functional.
  It is shown that the Hausdorff dimension d coincides with the abscissa of
convergence of the zeta function of |D|^(-1), and that the set of t's for which
there exists a singular trace tau_omega giving rise to a non-trivial functional
is an interval containing d. Moreover, the endpoints of such traceability
interval have a dimensional interpretation. The corresponding functionals are
called Hausdorff-Besicovitch functionals.
  These definitions are tested on fractals in R, by computing the mentioned
quantities and showing in many cases their correspondence with classical
objects. In particular, for self-similar fractals the traceability interval
consists only of the Hausdorff dimension, and the corresponding
Hausdorff-Besicovitch functional gives rise to the Hausdorff measure. More
generally, for any limit fractal, the described functionals do not depend on
the generalized limit omega.
