A classical inequality, which is known for families of monotone functions, is generalized to a larger class of families of measurable functions. Moreover we characterize all the families of functions for which the equality holds. We give two applications of this result, one of them to a problem arising from probability theory.

We show that there is a Bore1 measurable selection of Paretian utilities in ‘markets with a continuum of traders’. Theorem: Let T and X be Polish spaces, R a Bore1 subset of T x X xX, and B={(t,x): (t,x,x)ER}. Suppose that for each t, R,={(x,y): (t,x,y)ER} is a preference order on B, = {x: (t,x) EB). Then there is a Bore1 measurable function f: B-$0, l] such that for all t E IT; f,: B,+[O, I] i...