Almost everywhere domination and superhighness

13 years 3 months ago

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Let ω denote the set of natural numbers. For functions f, g : ω → ω, we say that f is dominated by g if f(n) < g(n) for all but ﬁnitely many n ∈ ω. We consider the standard “fair coin” probability measure on the space 2ω of inﬁnite sequences of 0’s and 1’s. A Turing oracle B is said to be almost everywhere dominating if, for measure one many X ∈ 2ω , each function which is Turing computable from X is dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of Kjos-Hanssen, Kjos-Hanssen/Miller/Solomon, and others concerning LR-reducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i.e., 0′′ is truth-table computable from B′ , the Turing j...

Stephen G. Simpson

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