# 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 $langlemu\_smid\; sin\{\}^\{\}y\; angle\; math>\; such\; that\; the\; following\; conditions\; hold:$

* $mu\_s$ is a countably-additive measure on $\{tmidlangle\; s,t\; anglein\; T\}$ .

* The measures are in some sense compatible under restriction of sequences: if $s\_1subseteq\; s\_2$, then $mu\_\{s\_1\}(X)=1iffmu\_\{s\_2\}(\{tmid\; tupharpoonright\; lh(s\_1)in\; X\})=1$.

* If $x$ is in the projection of $T$, the ultrapower by $langlemu\_\{xupharpoonright\; n\}mid\; ninomega\; angle$ is wellfounded.An equivalent definition is produced when the final condition is replaced with the following:

* There are $langlemu\_smid\; sin\{\}^omega\; Y\; angle$ such that if $x$ is in the projection of $[T]$ and $forall\; ninomega,mu\_\{xupharpoonright\; n\}(X\_n)=1$, then there is $fin\{\}^omega\; Z$ 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***

*Wikimedia Foundation.
2010.*

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