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Trivial Measures are not so Trivial

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Abstract

Although algorithmic randomness with respect to various non-uniform computable measures is well-studied, little attention has been paid to algorithmic randomness with respect to computable trivial measures, where a measure μ on 2ω is trivial if the support of μ consists of a countable collection of sequences. In this article, it is shown that there is much more structure to trivial computable measures than has been previously suspected.

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Notes

  1. In general, we can consider any convex sum \(\frac {\alpha \mu +(1-\alpha )\nu }{2}\) of μ and ν, as long as α∈[0,1] is computable.

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Acknowledgments

The author would like to thank Laurent Bienvenu, Peter Cholak, and Damir Dzhafarov for helpful conversations on the material in this article (which appeared in the author’s dissertation). The author would also like to thank Paul Shafer for useful feedback on a preliminary draft of this article.

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, Gainesville, FL, 32611, USA

    Christopher P. Porter

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  1. Christopher P. Porter
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Correspondence to Christopher P. Porter.

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The author was supported by the National Science Foundation under grant OISE-1159158.

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Porter, C.P. Trivial Measures are not so Trivial. Theory Comput Syst 56, 487–512 (2015). https://doi.org/10.1007/s00224-015-9614-8

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