Martin measure

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 ${\displaystyle X\leq _{T}Y}$;^[1] that is, the set of Turing degrees that are "at least as complex" as ${\displaystyle X}$ under Turing reduction. In order-theoretic terms, the cone of ${\displaystyle [X]}$ is the upper set of ${\displaystyle [X]}$.

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.^[1] It is similar to Wadge's lemma for Wadge degrees, and is important for the following result.

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.

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.

• Moschovakis, Yiannis N. (2009). Descriptive Set Theory. Mathematical surveys and monographs. Vol. 155 (2nd ed.). American Mathematical Society. p. 338. ISBN 9780821848135.