Social choice trade-offs for an arbitrary measure: With application to uncertain or fuzzy agenda

X is a finite set of alternatives that is to be socially ordered (as a function of individual preferences). We are given an arbitrary measure on X. If f is a transitive valued social welfare function that satisfies Arrowʹs independence of irrelevant alternatives condition, then for any fraction t, either some individual dictates on a subset of X of measure at least t, or else the set of ordered pairs of distinct alternatives that are socially ordered without consulting anyoneʹs preferences has a product measure 1−t. (The result can be extended to countably infinite X.)