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.
as the set of all Turing degrees
;[1] that is, the set of Turing degrees that are "at least as complex" as
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.
, 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.
It is easy to show that a countable intersection of cones is itself a cone; the Martin measure is therefore a countably complete filter.
is measurable under the axiom of determinacy.
This result shows part of the important connection between determinacy and large cardinals.