Martin measure

 ( Descriptive Set Theory ) 

In descriptive set theory, the Martin measure is a filter on the set of of sets of , named after Donald A. Martin. Under the axiom of determinacy it can be shown to be an ultrafilter.

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Definition Let $D$ be the set of Turing degrees of sets of natural numbers. Given some equivalence class $X\backslash in\; D$, we may define the cone (or upward cone) of $X$ as the set of all Turing degrees $Y$ such that $X\backslash le\_T\; Y$;D. Martin, H. G. Dales, Truth in Mathematics, ch. "Mathematical Evidence", p.223. Oxford Science Publications, 1998. that is, the set of Turing degrees that are "at least as complex" as $X$ under Turing reduction. In order-theoretic terms, the cone of $X$ is the upper set of $X$.

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. It is similar to Wadge hierarchy for Wadge degrees, and is important for the following result.

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.

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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 $\backslash omega\_1$ by a simple mapping, tells us that $\backslash omega\_1$ is measurable under the axiom of determinacy. This result shows part of the important connection between determinacy and .

• Yiannis N. Moschovakis (2025). 9780821848135, American Mathematical Society. . ISBN 9780821848135

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