×

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
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Located sets and reverse mathematics 65 pp 1451– (2000)
[2] Abhandlungen der Königlichen sächsischen Gesellschaft der Wissenschaften (Mathematisch-physische Klasse) 31 pp 296– (1909)
[3] DOI: 10.1007/BF01459088 · JFM 54.0056.06 · doi:10.1007/BF01459088
[4] DOI: 10.1007/BF01621469 · Zbl 0718.03043 · doi:10.1007/BF01621469
[5] DOI: 10.1215/S0012-7094-65-03247-3 · Zbl 0134.00805 · doi:10.1215/S0012-7094-65-03247-3
[6] Proceedings of the symposium on mathematical theory of automata (New York, 1962) pp 71– (1963)
[7] Subsystems of second order arithmetic (1999) · Zbl 0909.03048
[8] Degrees joining to 0’ 46 pp 714– (1981) · Zbl 0517.03014
[9] DOI: 10.1016/j.aim.2004.10.006 · Zbl 1141.03017 · doi:10.1016/j.aim.2004.10.006
[10] DOI: 10.1002/malq.19660120125 · Zbl 0181.30504 · doi:10.1002/malq.19660120125
[11] Transactions of the American Mathematical Society 173 pp 33– (1972) · Zbl 0247.00014
[12] DOI: 10.1007/BFb0076224 · doi:10.1007/BFb0076224
[13] DOI: 10.2140/pjm.1972.40.605 · Zbl 0209.02201 · doi:10.2140/pjm.1972.40.605
[14] Transactions of the American Mathematical Society 275 pp 599– (1983)
[15] Logic, methodology and philosophy of science, VIII (Moscow, 1987) 126 pp 191– (1989)
[16] Almost everywhere domination 69 pp 914– (2004)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.