Martin measure

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 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 $X\le_T Y$; that is, the set of Turing degrees which are "more complex" than 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 ω₁ by a simple mapping, tells us that ω₁ is measurable under the axiom of determinacy. This result shows part of the important connection between determinacy and large cardinals.

References

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.