  We continue the study of the relationship between Dixmier traces and
noncommutative residues initiated by A. Connes. The utility of the residue
approach to Dixmier traces is shown by a characterisation of the noncommutative
integral in Connes' noncommutative geometry (for a wide class of Dixmier
traces) as a generalised limit of vector states associated to the eigenvectors
of a compact operator (or an unbounded operator with compact resolvent), i.e.
as a generalised quantum limit. Using the characterisation, a criteria
involving the eigenvectors of a compact operator and the projections of a von
Neumann subalgebra of bounded operators is given so that the noncommutative
integral associated to the compact operator is normal, i.e. satisfies a
monotone convergence theorem, for the von Neumann subalgebra.
