Homogeneous tree

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.