## Notre Dame Journal of Formal Logic

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

### Randomness and Semimeasures

Laurent Bienvenu, Rupert Hölzl, Christopher P. Porter, and Paul Shafer

#### 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**

Received: 14 October 2013

Accepted: 23 September 2014

First available in Project Euclid: 15 March 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00294527-3839446

**Subjects**

Primary: 03D32: Algorithmic randomness and dimension [See also 68Q30]

**Keywords**

semimeasures measures algorithmic randomness randomness

#### 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