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 [math]\displaystyle{ D }[/math] be the set of Turing degrees of sets of natural numbers. Given some equivalence class [math]\displaystyle{ [X]\in D }[/math], we may define the cone (or upward cone) of [math]\displaystyle{ [X] }[/math] as the set of all Turing degrees [math]\displaystyle{ [Y] }[/math] such that [math]\displaystyle{ X\le_T Y }[/math]; that is, the set of Turing degrees that are "at least as complex" as [math]\displaystyle{ X }[/math] under Turing reduction. In order-theoretic terms, the cone of [math]\displaystyle{ [X] }[/math] is the upper set of [math]\displaystyle{ [X] }[/math].

Assuming the axiom of determinacy, the cone lemma states that if A is a set of Turing degrees, either A includes a cone or the complement of A contains a cone. It is similar to Wadge's lemma for Wadge degrees, and is important for the following result.

We say that a set [math]\displaystyle{ A }[/math] of Turing degrees has measure 1 under the Martin measure exactly when [math]\displaystyle{ A }[/math] contains some cone. Since it is possible, for any [math]\displaystyle{ A }[/math], to construct a game in which player I has a winning strategy exactly when [math]\displaystyle{ A }[/math] contains a cone and in which player II has a winning strategy exactly when the complement of [math]\displaystyle{ A }[/math] 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 [math]\displaystyle{ \omega_1 }[/math] by a simple mapping, tells us that [math]\displaystyle{ \omega_1 }[/math] 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. https://books.google.com/books?id=w0GZAwAAQBAJ&pg=PA338.