In descriptive set theory, a tree over a product set is said to be homogeneous if there is a system of measures such that the following conditions hold: is a countably-additive measure on . The measures are in some sense compatible under restriction of sequences: if , then . If is in the projection of , the ultrapower by is wellfounded.
An equivalent definition is produced when the final condition is replaced with the following: There are such that if is in the projection of and , then there is such that . This condition can be thought of as a sort of countable completeness condition on the system of measures.
is said to be -homogeneous if each is -complete.
Homogeneous trees are involved in Martin and Steel's proof of projective determinacy.
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