Английская Википедия:Homogeneous tree

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

• <math>\mu_s</math> is a countably-additive measure on <math>\{t\mid\langle s,t\rangle\in T\}</math> .
• The measures are in some sense compatible under restriction of sequences: if <math>s_1\subseteq s_2</math>, then <math>\mu_{s_1}(X)=1\iff\mu_{s_2}(\{t\mid t\upharpoonright lh(s_1)\in X\})=1</math>.
• If <math>x</math> is in the projection of <math>T</math>, the ultrapower by <math>\langle\mu_{x\upharpoonright n}\mid n\in\omega\rangle</math> is wellfounded.

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

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

<math>T</math> is said to be <math>\kappa</math>-homogeneous if each <math>\mu_s</math> is <math>\kappa</math>-complete.

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

References

• Шаблон:Cite journal

Шаблон:Settheory-stub