title: Tail-Measurable Functions and Their Corresponding Induced Classes, and Some Determinacy Conditions Involving 3-Player Games
Creators: name: Reagan, Joshua
Publisher: name: University of Nevada, Las Vegas
Year: 2021

description: In this dissertation, we have two main categories of results. The first is regarding certain point-classes, and the second is regarding 3-player games.

The point-classes of Baire Space, \mathcal{N}, in the Borel and Projective Hierarchies, as well as Hausdorff's Difference Hierarchy have been well studied, and there has been much research into further stratifying these hierarchies. One area of particular interest falls in between the point-classes \mathbf{\Pi}_\mathbf{1}^\mathbf{1} and \Delta\left(\omega^2-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}\right). It is well known that the point-classes \beta-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}, for \beta\in\omega^2, stratify this region of the projective hierarchy, with the point-class \bigcup_{\beta\in\omega^2}\beta-\mathbf{\Pi}_\mathbf{1}^\mathbf{1} still falling strictly below \Delta\left(\omega^2-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}\right). Dr. Derrick DuBose developed multiple point-classes, including \left(\kappa\ast\mathbf{\Pi}_\mathbf{1}^\mathbf{1}\right)^\ast for \kappa\in\omega_1. Using determinacy results, DuBose proved that certain of his point-classes further stratify the region between \bigcup_{\beta\in\omega^2}\beta-\mathbf{\Pi}_\mathbf{1}^\mathbf{1} and \Delta\left(\omega^2-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}\right).

In this dissertation, we define a new type of classification for functions, which we will refer to as \Gamma Tail-Measurable, as well as bounded \Gamma Tail-Measurable, where \Gamma is a point-class. We also define what we will mean for certain functions and certain sequences to be jointly bounded, that is to say bounded together. Using tail-measurable functions, we define a new manner in which to define certain point-classes of Baire space. When certain bounded tail-measurable functions are used, we will prove that the point-classes produced are exactly the point-classes developed by DuBose. We also will show that by using functions that are tail-measurable (but not bounded), we can produce point-classes that contain all of DuBose's point-classes that fall below \Delta\left(\omega^2-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}\right). Moreover, for certain sets X, defined from tail-measurable functions and sequences that are jointly unbounded, these point-classes contain every set A\subseteq X where A has cardinality at most \aleph_1.

Towards our goal, we review certain topological definitions including the definitions of the Borel and Projective Hierarchies, as well as Hausdorff's Difference Hierarchy. We also review some point-classes in the Projective Hierarchy developed by Dr. Derrick DuBose.

The study of determinacy of 2-player games on certain game trees is also an active area of research. While the most common game tree is the tree with height \omega and moves from \omega, there have been studies of the determinacy of 2-player games on other game trees, including trees of variable height. Many of the determinacy results use large cardinal hypotheses, such as ``0^# exists'', in order to calibrate the strength of the determinacy of certain point-classes in the Projective Hierarchy. It is well known that there exist games with 3 or more players that are not determined in which the payoff sets are of low complexity, e.g., clopen, in the Borel Hierarchy.

In this dissertation, we review some definitions concerning 2-player games and determinacy, and review some well-known determinacy results. We then adjust these definitions for 3-player games, and define what we will mean by imposing rules on these games. In effect, imposing a rule on a 3-player game amounts to changing the game tree on which the game is played. We then adjust Wolfe's proof of \Sigma_\mathbf{2}^\mathbf{0} determinacy for 2-player games to prove that 3-player games of a specific form are determined provided that a certain rule is imposed.

We also define a special class of 3-player games, which we will refer to as 3213-Games. We will explore some properties of these games, and will define rules that will yield the determinacy of these games.