Measurable operators and the asymptotics of heat kernels and zeta functions

In this note we answer some questions inspired by the introduction in Connes (1988, 1994) [6,7] of the notion of measurable operators using Dixmier traces. These questions concern the relationship of measurability to the asymptotics of ζ-functions and heat kernels. The answers have remained elusive for some 15 years.