Universal measurability and the Hochschild class of the Chern character

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.