A Game for the Borel Functions
Brian Thomas Semmes
Abstract:
In this thesis, we deal with classes of functions on Baire space. For
some important function classes, game representations are known and
proved to be very useful. The most prominent example is Wadge’s
characterization of the continuous func- tions that allowed the
development of the theory of the Wadge hierarchy; in 2006, based on a
result of Jayne and Rogers, Andretta gave a game representation for
the ∆1 functions (in the language of this thesis, this is the class
Λ2,2 ). Game characterizations are important as they allow for
“Wadge-style proof techniques”.
In their paper on Borel functions, Andretta and Martin lament that
“there is no analogue of the Wadge/Lipschitz games for Borel functions,
[and] hence many of the standard proofs for the Wadge hierarchy do not
generalize in a straightforward way to the Borel set-up.”
This suggested two important questions:
1. Can similar characterizations be given for other function
classes, most notably for the class of all Borel functions and the
class Λ3,3 ?
2. Is there an analogue of the Jayne-Rogers theorem at the third
level of the Borel hierarchy?
In this thesis, we give positive answers to these questions.
Keywords: