In descriptive set theory, a tree over a product set Y × Z is said to be homogeneous if there is a system of measures ⟨ μ s ∣ s ∈ < ω Y ⟩ such that the following conditions hold:
μ s is a countably-additive measure on { t ∣ ⟨ s , t ⟩ ∈ T } .The measures are in some sense compatible under restriction of sequences: if s 1 ⊆ s 2 , then μ s 1 ( X ) = 1 ⟺ μ s 2 ( { t ∣ t ↾ l h ( s 1 ) ∈ X } ) = 1 .If x is in the projection of T , the ultrapower by ⟨ μ x ↾ n ∣ n ∈ ω ⟩ is wellfounded.An equivalent definition is produced when the final condition is replaced with the following:
There are ⟨ μ s ∣ s ∈ ω Y ⟩ such that if x is in the projection of [ T ] and ∀ n ∈ ω μ x ↾ n ( X n ) = 1 , then there is f ∈ ω Z such that ∀ n ∈ ω f ↾ n ∈ 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 κ -homogeneous if each μ s is κ -complete.
Homogeneous trees are involved in Martin and Steel's proof of projective determinacy.
