Rearrangements and polar factorisation of countably degenerate functions

This paper proves some extensions of Brenier's theorem that an integrable vector-valued function u, satisfying a nondegeneracy condition, admits a unique polar factorisation u = u# ° s. Here u# is the monotone rearrangement of u, equal to the gradient of a convex function almost everywhere on a bounded connected open set Y with smooth boundary, and s is a measure-preserving mapping. We show that two weaker alternative hypotheses are sufficient for the existence of the factorisation; that u# be almost injective (in which case s is unique), or that u be countably degenerate (which allows u to have level sets of positive measure). We allow Y to be any set of finite positive Lebesgue measure. Our construction of the measure-preserving map s is especially simple.