Martin measure

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In descriptive set theory, the Martin measure is a filter on the set of Turing degrees of sets of natural numbers. 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

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