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 $\{tmidlangle\; s,t\; anglein\; T\}$ .
* 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

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