## Notre Dame Journal of Formal Logic

### Randomness and Semimeasures

#### Abstract

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.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 58, Number 3 (2017), 301-328.

Dates
Accepted: 23 September 2014
First available in Project Euclid: 15 March 2017

https://projecteuclid.org/euclid.ndjfl/1489543214

Digital Object Identifier
doi:10.1215/00294527-3839446

Mathematical Reviews number (MathSciNet)
MR3681097

Zentralblatt MATH identifier
06761611

Subjects

#### Citation

Bienvenu, Laurent; Hölzl, Rupert; Porter, Christopher P.; Shafer, Paul. Randomness and Semimeasures. Notre Dame J. Formal Logic 58 (2017), no. 3, 301--328. doi:10.1215/00294527-3839446. https://projecteuclid.org/euclid.ndjfl/1489543214

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