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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.