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. 2 (1). Journal of the American Mathematical Society, Vol. 2, No. 1: 71–125. doi:10.2307/1990913. JSTOR 1990913.