Open Access
2017 Randomness and Semimeasures
Laurent Bienvenu, Rupert Hölzl, Christopher P. Porter, Paul Shafer
Notre Dame J. Formal Logic 58(3): 301-328 (2017). DOI: 10.1215/00294527-3839446


A semimeasure is a generalization of a probability measure obtained by relaxing the additivity requirement to superadditivity. We introduce and study several randomness notions for left-c.e. semimeasures, a natural class of effectively approximable semimeasures induced by Turing functionals. Among the randomness notions we consider, the generalization of weak 2-randomness to left-c.e. semimeasures is the most compelling, as it best reflects Martin-Löf randomness with respect to a computable measure. Additionally, we analyze a question of Shen, a positive answer to which would also have yielded a reasonable randomness notion for left-c.e. semimeasures. Unfortunately, though, we find a negative answer, except for some special cases.


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Laurent Bienvenu. Rupert Hölzl. Christopher P. Porter. Paul Shafer. "Randomness and Semimeasures." Notre Dame J. Formal Logic 58 (3) 301 - 328, 2017.


Received: 14 October 2013; Accepted: 23 September 2014; Published: 2017
First available in Project Euclid: 15 March 2017

zbMATH: 06761611
MathSciNet: MR3681097
Digital Object Identifier: 10.1215/00294527-3839446

Primary: 03D32

Keywords: algorithmic randomness , Measures , randomness , semimeasures

Rights: Copyright © 2017 University of Notre Dame

Vol.58 • No. 3 • 2017
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