Homogeneous tree
# Homogeneous tree

In descriptive set theory, a tree over a product set $Y\times Z$ is said to be homogeneous if there is a system of measures $\langle \mu _{s}\mid s\in^{<\omega }Y\rangle$ such that the following conditions hold:

* $\mu _{s}$ is a countably-additive measure on $\{t\mid \langle s,t\rangle \in T\}$ .
* The measures are in some sense compatible under restriction of sequences: if $s_{1}\subseteq s_{2}$, then $\mu _{s_{1}}(X)=1\iff \mu _{s_{2}}(\{t\mid t\upharpoonright lh(s_{1})\in X\})=1$.
* If $x$ is in the projection of $T$, the ultrapower by $\langle \mu _{x\upharpoonright n}\mid n\in \omega \rangle$ is wellfounded.

An equivalent definition is produced when the final condition is replaced with the following:

$T$ is said to be $\kappa$-homogeneous if each $\mu _{s}$ is $\kappa$-complete.

Homogeneous trees are involved in Martin and Steel's proof of projective determinacy.

