Lebesgue measure and social choice trade-offs

An Arrovian social choice rule is a social welfare function satisfying independence of irrelevant alternatives and transitivity of social preference. Assume a measurable outcome space X with its (Lebesgue) measure normalized to unity. For any Arrovian rule and any fraction t, either some individual dictates over a subset of X of measure t or more, or at least a fraction 1-t of the pairs of distinct alternatives have their social ordering fixed independently of individual preferences. Also, for any positive integer β (less than the total number of individuals), there is some subset H of society consisting of all but β persons such that the fraction of outcome pairs (x, y) that are social ranked without consulting the preferences of anyone in H, whenever no individual is indifferent between x and y, is at least 1-1/4 β.