We unify various constructions and contribute to the theory of singular symmetric functionals on Marcinkiewicz function/operator spaces. This affords a new approach to the non-normal Dixmier and Connes-Dixmier traces (introduced by Dixmier and adapted to non-commutative geometry by Connes) living on a general Marcinkiewicz space associated with an arbitrary semifinite von Neumann algebra. The corollaries to our approach, stated in terms of the oper-is a special example of an operator Marcinkiewicz space), are: (i) a new characterization of the set of all positive measurable operators from L (1,∞) , i.e. those on which an arbitrary Connes-Dixmier trace yields the same value. In the special case, when the operator ideal L (1,∞) is considered on a type I infinite factor, a bounded operator x belongs to L (1,∞) if and only if the sequence of singular numbers {s n (x)} n 1 (in the descending order and counting the multiplicities) satisfies x (1,∞) := sup N 1 1 Log(1+N) N n=1 s n (x) < ∞. In this case, our characterization amounts to saying that a positive element x ∈ L (1,∞) is measurable if and only if lim N→∞ 1 Log N N n=1 s n (x) exists; (ii) the set of Dixmier traces and the * Corresponding author. Fax: +61 882012904. E-mail addresses: sed@vmail.ru (A. Sedaev), sukochev@infoeng.flinders.edu.au (F.A. Sukochev).