Homogeneous tree

In descriptive set theory, a tree over a product set $Y\times Z$ is said to be homogeneous if there is a system of measures $\langle \mu _{s}\mid s\in {}^{{<\omega }}Y\rangle$ such that the following conditions hold:

$\mu _{s}$ is a countably-additive measure on $\{t\mid \langle s,t\rangle \in T\}$ .
The measures are in some sense compatible under restriction of sequences: if $s_{1}\subseteq s_{2}$ , then $\mu _{{s_{1}}}(X)=1\iff \mu _{{s_{2}}}(\{t\mid t\upharpoonright lh(s_{1})\in X\})=1$ .
If $x$ is in the projection of $T$ , the ultrapower by $\langle \mu _{{x\upharpoonright n}}\mid n\in \omega \rangle$ is wellfounded.

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

There are $\langle \mu _{s}\mid s\in {}^{\omega }Y\rangle$ such that if $x$ is in the projection of $[T]$ and $\forall n\in \omega \,\mu _{{x\upharpoonright n}}(X_{n})=1$ , then there is $f\in {}^{\omega }Z$ such that $\forall n\in \omega \,f\upharpoonright n\in X_{n}$ . 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

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. JSTOR 1990913.

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