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.

DefinitionEdit

Let   be the set of Turing degrees of sets of natural numbers. Given some equivalence class  , we may define the cone (or upward cone) of   as the set of all Turing degrees   such that  ; that is, the set of Turing degrees that are "at least as complex" as   under Turing reduction.

We say that a set   of Turing degrees has measure 1 under the Martin measure exactly when   contains some cone. Since it is possible, for any  , to construct a game in which player I has a winning strategy exactly when   contains a cone and in which player II has a winning strategy exactly when the complement of   contains a cone, the axiom of determinacy implies that the measure-1 sets of Turing degrees form an ultrafilter.

ConsequencesEdit

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   by a simple mapping, tells us that   is measurable under the axiom of determinacy. This result shows part of the important connection between determinacy and large cardinals.

ReferencesEdit

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