  Given a C*-algebra A with a semicontinuous semifinite trace tau acting on the
Hilbert space H, we define the family R of bounded Riemann measurable elements
w.r.t. tau as a suitable closure, a la Dedekind, of A, in analogy with one of
the classical characterizations of Riemann measurable functions, and show that
R is a C*-algebra, and tau extends to a semicontinuous semifinite trace on R.
  Then, unbounded Riemann measurable operators are defined as the closed
operators on H which are affiliated to A'' and can be approximated in measure
by operators in R, in analogy with unbounded Riemann integration. Unbounded
Riemann measurable operators form a tau-a.e. bimodule on R, denoted by R^, and
such bimodule contains the functional calculi of selfadjoint elements of R
under unbounded Riemann measurable functions. Besides, tau extends to a
bimodule trace on R^.
  Type II_1 singular traces for C*-algebras can be defined on the bimodule of
unbounded Riemann-measurable operators. Noncommutative Riemann integration, and
singular traces for C*-algebras, are then used to define Novikov-Shubin numbers
for amenable open manifolds, show their invariance under quasi-isometries, and
prove that they are (noncommutative) asymptotic dimensions.
