  To any spectral triple (A,D,H) a dimension d is associated, in analogy with
the Hausdorff dimension for metric spaces. Indeed d is the unique number, if
any, such that |D|^-d has non trivial logarithmic Dixmier trace. Moreover, when
d is finite non-zero, there always exists a singular trace which is finite
nonzero on |D|^-d, giving rise to a noncommutative integration on A.
  Such results are applied to fractals in R, using Connes' spectral triple, and
to limit fractals in R^n, a class which generalises self-similar fractals,
using a new spectral triple. The noncommutative dimension or measure can be
computed in some cases. They are shown to coincide with the (classical)
Hausdorff dimension and measure in the case of self-similar fractals.
