Homogeneous tree

In descriptive set theory, a tree over a product set [math]\displaystyle{ Y\times Z }[/math] is said to be homogeneous if there is a system of measures [math]\displaystyle{ \langle\mu_s\mid s\in{}^{\lt \omega}Y\rangle }[/math] such that the following conditions hold:

• [math]\displaystyle{ \mu_s }[/math] is a countably-additive measure on [math]\displaystyle{ \{t\mid\langle s,t\rangle\in T\} }[/math] .
• The measures are in some sense compatible under restriction of sequences: if [math]\displaystyle{ s_1\subseteq s_2 }[/math], then [math]\displaystyle{ \mu_{s_1}(X)=1\iff\mu_{s_2}(\{t\mid t\upharpoonright lh(s_1)\in X\})=1 }[/math].
• If [math]\displaystyle{ x }[/math] is in the projection of [math]\displaystyle{ T }[/math], the ultrapower by [math]\displaystyle{ \langle\mu_{x\upharpoonright n}\mid n\in\omega\rangle }[/math] is wellfounded.

An equivalent definition is produced when the final condition is replaced with the following:

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

[math]\displaystyle{ T }[/math] is said to be [math]\displaystyle{ \kappa }[/math]-homogeneous if each [math]\displaystyle{ \mu_s }[/math] is [math]\displaystyle{ \kappa }[/math]-complete.

Homogeneous trees are involved in Martin and Steel's proof of projective determinacy.

References

• Martin, Donald A. and John R. Steel (Jan 1989). "A Proof of Projective Determinacy". Journal of the American Mathematical Society (Journal of the American Mathematical Society, Vol. 2, No. 1) 2 (1): 71–125. doi:10.2307/1990913. 

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