Constructive approach to the monotone rearrangement of functions

We detail a simple procedure (easily convertible to an algorithm) for constructing, from quasi-uniform samples of f , a sequence of linear spline functions converging to the monotone rearrangement of f , in the case where f is an almost everywhere continuous function defined on a bounded set 12 with negligible boundary. Under additional assumptions on f and 12, we prove that the convergence of the sequence is uniform. We also show that the same procedure applies to arbitrary measurable functions too, but with the substantial difference that in this case the procedure has only a theoretical interest and cannot be converted to an algorithm.⃝c 2021 Elsevier GmbH. All rights reserved.