Homogeneously Suslin set

In descriptive set theory, a set $$S$$ is said to be homogeneously Suslin if it is the projection of a homogeneous tree. $$S$$ is said to be $$\kappa$$-homogeneously Suslin if it is the projection of a $$\kappa$$-homogeneous tree.

If $$A\subseteq{}^\omega\omega$$ is a $$\mathbf{\Pi}_1^1$$ set and $$\kappa$$ is a measurable cardinal, then $$A$$ is $$\kappa$$-homogeneously Suslin. This result is important in the proof that the existence of a measurable cardinal implies that $$\mathbf{\Pi}_1^1$$ sets are determined.