  We study notions of measurability for singular traces, and characterise
universal measurability for operators in Dixmier ideals. This measurability
result is then applied to improve on the various proofs of Connes'
identification of the Hochschild class of the Chern character of Dixmier
summable spectral triples.
  The measurability results show that the identification of the Hochschild
class is independent of the choice of singular trace. As a corollary we obtain
strong information on the asymptotics of the eigenvalues of operators naturally
associated to spectral triples (A,H,D) and Hochschild cycles for A.
