Existence and uniqueness of monotone measure-preserving maps

Introduction. Given a pair of Borel probability measures # and v on Rd, it is natural to ask whether v can be obtained from # by redistributing its mass in a canonical way. In the case of the line d 1 the answer is clear: as long as both measures are free from atoms--#x vx 0mthere is a map y (x) of the line to itself for which Uniquely determined /-almost everywhere, this map may be taken to be nondecreasing by a suitable choice of y (x) e Rw __ₒ at the remaining points. Interpreting # and v as the initial and final distribution of a one-dimensional fluid, the transformation y (x) gives a rearrangement of fluid particles yielding final state v from the initial state #; this rearrangement is characterized by the fact that it preserves particle ordering, obviating any need for two particles to cross. Although the generalization of this construction to higher dimensions is the focus of this paper, the one-dimensional case will be pursued slightly further: when the measures are absolutely continuous with respect to Lebesgue--d# (x) = f (x) dx and dr (y) g (y) dymthen, formally at least (neglecting regularity issues), the fundamental theorem of calculus yields