# Homogeneous tree

In descriptive set theory, a tree over a product set $Y imes Z$ is said to be homogeneous if there is a system of measures
* $mu_s$ is a countably-additive measure on $\left\{tmidlangle s,t anglein T\right\}$ .
* The measures are in some sense compatible under restriction of sequences: if $s_1subseteq s_2$, then $mu_\left\{s_1\right\}\left(X\right)=1iffmu_\left\{s_2\right\}\left(\left\{tmid tupharpoonright lh\left(s_1\right)in X\right\}\right)=1$.
* If $x$ is in the projection of $T$, the ultrapower by $langlemu_\left\{xupharpoonright n\right\}mid ninomega angle$ is wellfounded.

An equivalent definition is produced when the final condition is replaced with the following:
* There are $langlemu_smid sin\left\{\right\}^omega Y angle$ such that if $x$ is in the projection of $\left[T\right]$ and $forall ninomega,mu_\left\{xupharpoonright n\right\}\left(X_n\right)=1$, then there is $fin\left\{\right\}^omega Z$ such that $forall ninomega,fupharpoonright nin X_n$. 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.

References

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