Homogeneous tree explained

In descriptive set theory, a tree over a product set

Y x 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} \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} (X _{n)=1} , then there is f\in{} ^{\omega} Z such that \forall n\in\omegaf\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. 71–125. 10.2307/1990913. Journal of the American Mathematical Society, Vol. 2, No. 1. 1990913. free.