A random variable can be equivalently regarded to as a function or as a “set”, namely, that of the points lying below (when positive) and above (when negative) its graph. The second approach, proposed by Segal in 1989, is known as the
measure (or measurement) representation approach
. On a technical ground, it allows for using Measure theory tools instead of Functional analysis ones, thus making often possible to reach new and deeper conclusions. On an interpretative ground, it makes clear how expectation and expected utility, either classical or
Choquet, are structurally analogous and, moreover, it allows for dealing with new and more general types of expectations including, e.g., state dependence.
Starting from the measurement approach and from a decision-theoretical result by Castagnoli and LiCalzi (2006), we present a new representation theorem in the same perspective and, finally, we propose a definition of generalised expectations and two different concepts of associativity that can be imposed to them.