Homogeneous tree

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

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

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

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

${\displaystyle T}$ is said to be ${\displaystyle \kappa }$-homogeneous if each ${\displaystyle \mu _{s}}$ is ${\displaystyle \kappa }$-complete.

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

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