Homogeneous tree

In descriptive set theory, a tree over a product set Y×Z is said to be homogeneous if there is a system of measures ⟨μs∣s∈<ωY⟩ such that the following conditions hold:

* μs is a countably-additive measure on {t∣⟨s,t⟩∈T} .
* The measures are in some sense compatible under restriction of sequences: if s1⊆s2, then μs1(X)=1⟺μs2({t∣t↾lh(s1)∈X})=1.
* If x is in the projection of T, the ultrapower by ⟨μx↾n∣n∈ω⟩ is wellfounded.

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

* There are ⟨μs∣s∈ωY⟩ such that if x is in the projection of [T] and ∀n∈ωμx↾n(Xn)=1, then there is f∈ωZ such that ∀n∈ωf↾n∈Xn. This condition can be thought of as a sort of countable completeness condition on the system of measures.

T is said to be κ-homogeneous if each μs is κ-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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