# Martin measure

In descriptive set theory, the Martin measure is a filter on the set of Turing degrees of sets of natural numbers. Under the axiom of determinacy it can be shown to be an ultrafilter.

## Definition

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 ${\displaystyle X\leq _{T}Y}$; that is, the set of Turing degrees which are "more complex" than ${\displaystyle X}$ under Turing reduction.

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.

## 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 ${\displaystyle \omega _{1}}$ 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.

## References

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