Unraveling ${\mathbf\Pi}^1_1$ sets

Itay Neeman

Abstract:

We construct coverings which unravel ${\mathbf\Pi}^1_1$

sets. This in turn is used to prove, from optimal large

cardinal assumptions, the determinacy of games with

${\cal A}{\mathbf\Pi}^1_1$ payoff and the determinacy

of games with payoff in the $\sigma$ algebra generated

by ${\mathbf\Pi}^1_1$ sets.