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Uniform almost everywhere domination. (English) Zbl 1109.03034

Summary: We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for \(G_\delta\) sets. Our constructions essentially settle the reverse mathematical classification of this principle.

MSC:

03D25 Recursively (computably) enumerable sets and degrees
03F35 Second- and higher-order arithmetic and fragments
28E15 Other connections with logic and set theory
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