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{}^{ 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

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