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Martin measure

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

We say that a set ${\displaystyle A}$  of Turing degrees has measure 1 under the Martin measure exactly when ${\displaystyle A}$  contains some cone. Since it is possible, for any ${\displaystyle A}$ , to construct a game in which player I has a winning strategy exactly when ${\displaystyle A}$  contains a cone and in which player II has a winning strategy exactly when the complement of ${\displaystyle A}$  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 ${\displaystyle \omega _{1}}$  by a simple mapping, tells us that ${\displaystyle \omega _{1}}$  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. Vol. 155 (2nd ed.). American Mathematical Society. p. 338. ISBN 9780821848135.