In the framework of non-commutative geometry the singular (Dixmier) traces, originally introduced by J. Dixmier in 1966, have become an indispensable tool. These traces are defined via dilation invariant extended limits on the space of bounded measurable functions. Important results in non-commutative geometry (e.g. Connes Character Theorem, relation between Dixmier traces, heat functionals and zeta-function residues) are proved under various additional conditions on these extended limits.
Every such condition distinguishes a subclass of Dixmier traces. In the present talk we discuss the relation between these classes and an important concept of measurable operators (with respect to these subclasses) introduced in non-commutative geometry by A. Connes in 1988. In most cases we provide new characterisations of measurability and definitive description of classes of measurable operators.