Abstract:

Work throughout in the space ${2}^{\mathbb N}$ of infinite sequences of zeros and ones. (With the product topology this space is isomorphic to the cantor subset of the interval $[0,1]$.) Let $A\subset {2}^{\mathbb N}$ be given. The infinite game $G(A)$ is played as follows: $$\begin{array}{c|ccccccccccc}I & a_0 & & a_2 & & a_4 & & \dots \\ \hlineII & & a_1 & & a_3 & & \dots\end{array}$$Players I and II alternate playing $a_n\in \{ 0,1\}$, producing together the sequence $\vec{a}=\$. $\vec{a}$ is called a {\bf run} of the game $G(A)$. A run $\vec{a}$ is won by player I iff $\vec{a}\in A$. Otherwise the run is won by player II.The notion of a {\bf winning strategy} (for I or for II) is defined in the natural way. $G(A)$ is said to be {\bf determined} if one of the players has a winning strategy. One may think of the determinacy of $G(A)$ as stating that either:\begin{itemize}\item[] $\exists a_0\: \forall a_1\: \exists a_2 \:\forall a_3\: \dots\dots \:\: \\in A$; or else\item[] $\forall a_0\:\exists a_1\:\forall a_2\:\exists a_3\:\dots\dots \:\: \\not\in A$.\end{itemize}${\sf AD}(\Gamma)$, for a collection $\Gamma$ of subsets of $2^{\mathbb N}$, is the assertion that $G(A)$ is determined for all $A\in \Gamma$. Such assertions turn out to have a crucial role in the study of {\em definable} subsets of $2^{\mathbb N}$. \medskipIn this talk I will present some basic arguments which involve infinite games. I will mention infinite games of {\em imperfect} information and the assertion $\mbox{\sf Bl-AD}(\Gamma)$ that such games are determined. I will mention the relation between $\mbox{\sf Bl-AD}(\Gamma)$ and ${\sf AD}(\Gamma)$ (due to Martin in one direction, and to Martin--Neeman--Vervoort in the other). I will sketch a related argument, assuming basic closure properties on $\Gamma$, that ${\sf AD}(\Gamma)$ implies that all sets in $\Gamma$ are Lebesgue measurable. (The result is due to Mycielski--Swierczkowski. The argument I will sketch is due to Martin.)