Abstract
We study the uniform computational content of the Vitali Covering Theorem for intervals using the tool of Weihrauch reducibility. We show that a more detailed picture emerges than what a related study by Giusto, Brown, and Simpson has revealed in the setting of reverse mathematics. In particular, different formulations of the Vitali Covering Theorem turn out to have different uniform computational content. These versions are either computable or closely related to uniform variants of Weak Weak Kőnig’s Lemma.
This article is dedicated to Rod Downey on the occasion of his sixtieth birthday.
Vasco Brattka is supported by the National Research Foundation of South Africa.
Rupert Hölzl was partly supported by the Ministry of Education of Singapore through grant R146-000-184-112 (MOE2013-T2-1-062).
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Notes
- 1.
There is a slight ambiguity here, as we need to deal with open sets ranging beyond \({[0,1]}\). We shall understand these to be small enough in the sense that we cut away everything from a certain distance \(\varepsilon _n\) on. The exact constraints that these values \(\varepsilon _n\) need to satisfy are given in the proof.
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Brattka, V., Gherardi, G., Hölzl, R., Pauly, A. (2017). The Vitali Covering Theorem in the Weihrauch Lattice. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_14
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