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## Summary

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 $D$  be the set of Turing degrees of sets of natural numbers. Given some equivalence class $[X]\in D$ , we may define the cone (or upward cone) of $[X]$  as the set of all Turing degrees $[Y]$  such that $X\leq _{T}Y$ ; that is, the set of Turing degrees that are "at least as complex" as $X$  under Turing reduction.

We say that a set $A$  of Turing degrees has measure 1 under the Martin measure exactly when $A$  contains some cone. Since it is possible, for any $A$ , to construct a game in which player I has a winning strategy exactly when $A$  contains a cone and in which player II has a winning strategy exactly when the complement of $A$  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 $\omega _{1}$  by a simple mapping, tells us that $\omega _{1}$  is measurable under the axiom of determinacy. This result shows part of the important connection between determinacy and large cardinals.