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.

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.

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