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:
 * There are $$\langle\mu_s\mid s\in{}^\omega Y\rangle$$ such that if $$x$$ is in the projection of $$[T]$$ and $$\forall n\in\omega\,\mu_{x\upharpoonright n}(X_n)=1$$, then there is $$f\in{}^\omega Z$$ such that $$\forall n\in\omega\,f\upharpoonright n\in X_n$$. This condition can be thought of as a sort of countable completeness condition on the system of measures.

$$T$$ 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.