# Martin measure

In descriptive set theory, the **Martin measure** is a filter on the set of Turing degrees of sets of natural numbers, named after Donald A. Martin. Under the axiom of determinacy it can be shown to be an ultrafilter.

## Definition

Let be the set of Turing degrees of sets of natural numbers. Given some equivalence class , we may define the *cone* (or *upward cone*) of as the set of all Turing degrees such that ; that is, the set of Turing degrees which are "more complex" than under Turing reduction.

We say that a set of Turing degrees has measure 1 under the Martin measure exactly when contains some cone. Since it is possible, for any , to construct a game in which player I has a winning strategy exactly when contains a cone and in which player II has a winning strategy exactly when the complement of contains a cone, the axiom of determinacy implies that the measure-1 sets of Turing degrees form an ultrafilter.

## Consequences

It is easy to show that a countable intersection of cones is itself a cone; the Martin measure is therefore a countably complete filter. This fact, combined with the fact that the Martin measure may be transferred to by a simple mapping, tells us that is measurable under the axiom of determinacy. This result shows part of the important connection between determinacy and large cardinals.

## References

- Moschovakis, Yiannis N. (2009).
*Descriptive Set Theory*. Mathematical surveys and monographs.**155**(2nd ed.). American Mathematical Society. p. 338. ISBN 9780821848135.