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 formula_1 be the set of Turing degrees of sets of natural numbers. Given some equivalence class formula_2, we may define the "cone" (or "upward cone") of formula_3 as the set of all Turing degrees formula_4 such that formula_5; that is, the set of Turing degrees that are "at least as complex" as formula_6 under Turing reduction. We  say that a set formula_7 of Turing degrees has measure 1 under the Martin measure exactly when formula_7 contains some cone. Since it is possible, for any formula_7, to construct a game in which player I has a winning strategy exactly when formula_7 contains a cone and in which player II has a winning strategy exactly when the complement of formula_7 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 formula_12 by a simple mapping, tells us that formula_12 is measurable under the axiom of determinacy. This result shows part of the important connection between determinacy and large cardinals.