Higher Kurtz randomness

Abstract

A real $x$ is ${\Delta }_1^1$-Kurtz random (${\Pi }_1^1$-Kurtz random) if it is in no closed null ${\Delta }_1^1$ set (${\Pi }_1^1$ set). We show that there is a cone of ${\Pi }_1^1$-Kurtz random hyperdegrees. We characterize lowness for ${\Delta }_1^1$-Kurtz randomness as being ${\Delta }_1^1$-dominated and ${\Delta }_1^1$-semi-traceable.