Certain games, category, and measure

Abstract:

For $A \subset {2^\omega }$ and $X \subset \omega$ consider an infinite game $\Gamma (A,X)$ in which two players I and II choose ${c_n} \in \{ 0,1\}$. ${c_n}$ is chosen by I if $n \in X$ and by II if $n \in \omega \backslash X$. I wins if $({c_0},{c_1},{c_2}, \ldots ) \in A$. We analyze connections between $A$ and the family of all sets $X \subset \omega$ for which I has a winning strategy in $\Gamma (A,X)$. Certain similarities and differences appear if one formulates these connections in the language of category and of Lebesgue measure.