  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 non-trivial 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.
