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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).

In an increasingly globalized and interconnected world, where social and environmental change occur ever more rapidly, careful futures-oriented thinking becomes crucial for effective decision making. Foresight activities, including scenario development, quantitative modeling, and scenario-guided design of policies and programs, play a key role in exploring options to address socioeconomic and environmental challenges across many sectors and decision-making levels. We take stock of recent methodological developments in scenario and foresight exercises, seek to provide greater clarity on the many diverse approaches employed, and examine their use by decision makers in different fields and at different geographic, administrative, and temporal scales. Experience shows the importance of clearly formulated questions, structured dialog, carefully designed scenarios, sophisticated biophysical and socioeconomic analysis, and iteration as needed to more effectively link the growing scenarios and foresight community with today's decision makers and to better address the social, economic, and environmental challenges of tomorrow.