Martin’s Conjecture is a proposed classification of the definable functions on the Turing degrees. It is usually divided into two parts, the first of which classifies functions which are not above the identity and the second of which classifies functions which are above the identity. Slaman and Steel proved the second part of the conjecture for Borel functions which are order-preserving (i.e. which preserve Turing reducibility). We prove the first part of the conjecture for all order-preserving functions. We do this by introducing a class of functions on the Turing degrees which we call “measure-preserving” and proving that part 1 of Martin’s Conjecture holds for all measure-preserving functions and also that all nontrivial order-preserving functions are measure-preserving. Our result on measure-preserving functions has several other consequences for Martin’s Conjecture, including an equivalence between part 1 of the conjecture and a statement about the structure of the Rudin-Keisler order on ultrafilters on the Turing degrees.

马丁猜想是对图灵度上可定义函数的提议分类。它通常分为两部分，第一部分对不高于恒等的函数进行分类，第二部分对高于恒等的函数进行分类。 Slaman 和 Steel 证明了保序的 Borel 函数猜想的第二部分（即保留图灵可约性）。我们证明了所有保序函数猜想的第一部分。为此，我们引入一类关于图灵度的函数，我们称之为“测度保持”，并证明马丁猜想的第一部分适用于所有测度保持函数，并且所有非平凡的保序函数都是测度保持的。我们关于保测函数的结果对马丁猜想还有其他几个影响，包括猜想的第一部分与关于图灵度超滤器 Rudin-Keisler 阶结构的陈述之间的等价性。