## A computable topological space of measures

The content of the paper will be merged into Algorithmic randomness over general spaces.

News
June 13, 2011, Rejected
Sep 21, 2010, Submitted to a Journal

Title
A computable topological space of measures

Type
Fullpaper

Journal
submitted

Japanese summary

Abstract
We show that the space of bounded non-negative Borel measures on
a computable topological space is also a computable topological space
with A-topology. Then we de ne computable measures which may not
be probabilistic and may be even in nite. We also study randomness for
non-negative Borel measures which may not be probabilistic.

## Truth-table Schnorr randomness and truth-table reducible randomness

News
Mar 19, 2011. Added citation information
Mar 7, 2011. Published online
May 18, 2010. The paper “Truth-table Schnorr randomness and truth-table reducible randomness” is accepted by Mathematical Logic Quarterly on May 18, 2010.

Title
Truth-table Schnorr randomness and truth-table reducible randomness

Type
Fullpaper

Journal
Mathematical Logic Quarterly 57(3):323-338, 2011
DOI 10.1002/malq.200910128
Journal Page

preprint
Japanese summary

Abstract
Schnorr randomness and computably randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness (defined by Franklin and Stephan too only by martingales) and truth-table reducible randomness, for which we prove that van Lambalgen’s Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen’s Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness.

Cited by
@article{franklin2009van,
title={{van Lambalgen’s Theorem and high degrees}},
author={Franklin, J.N.Y. and Stephan, F.},
year={2009},
publisher={Submitted}
}
@misc{bienvenucharacterizing,
title={Characterizing lowness for Demuth randomness},
author={Bienvenu, L. and Downey, R. and Greenberg, N. and Nies, A. and Turetsky, D.},
publisher={Submitted}
}

## Generalizing randomness

News
Feb, 2011, Published
Dec 13, 2010, Submitted

Title
Generalizing randomness

Type
Survey

Journal
RIMS Kokyuroku
No. 1729
“Formal Systems and Computability Theory” (2010/09/13-2010/09-17)
pp. 84-94

preprint

## A Random Sequence of Reals

Since this paper was rejected by a Journal, I will divide the results, extend them and publish by some papers.

News
Jan 11, 2011, Rejected
Aug 2, 2010, Submitted to a Journal

Title
A Random Sequence of Reals

Type
Fullpaper

Journal
Unpublished

preprint
Japanese summary

Abstract
We define a random sequence of reals as a random point on a computable topological space. This randomness has three equivalent simple characterizations, namely, by tests, by martingales and by complexity. We prove that members of a random sequence are relatively random. Conversely a relatively random sequence of reals has a random sequence such that each corresponding member is Turing equivalent. Furthermore strong law of large numbers and the law of the iterated logarithm hold for each random sequence.

## An extension of van Lambalgen’s Theorem to infinitely many relative 1-random reals

Title
An extension of van Lambalgen’s Theorem to infinitely many relative 1-random reals

Type
Fullpaper

Journal
Notre Dame Journal of Formal Logic, 51(3):337-349, 2010.
Permanent link to this document in this Journal.
Received November 30, 2009; accepted December 8, 2009; printed June 16, 2010

preprint
Japanese summary

Abstract
Van Lambalgen’s Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen’s Theorem by considerint the join of infinitely many reals which are random relative to each other.
In addition, we study computability of the reals in the range of Omega operators. It is known that $\Omega^{\phi’}$ is high. We extend this result to that $\Omega^{\phi^{(n)}}$ is $\textrm{high}_n$. We also prove that there exists A such that, for each n, the real $\Omega^A_M$ is highn for some universal Turing machine M by using the extended van Lambalgen’s Theorem.

Cited by
@article{bienvenu2010ergodic,
title={{Ergodic-type characterizations of algorithmic randomness}},
author={Bienvenu, L. and Day, A. and Mezhirov, I. and Shen, A.},
journal={Programs, Proofs, Processes},
pages={49–58},
year={2010},
publisher={Springer}
}
@article{bienvenu2011constructive,
title={A constructive version of Birkhoffʼs ergodic theorem for Martin–L{\”o}f random points},
author={Bienvenu, L. and Day, A. and Hoyrup, M. and Mezhirov, I. and Shen, A.},
journal={Information and Computation},
year={2011},
publisher={Elsevier}
}
@article{yu2011characterizing,
title={Characterizing strong randomness via Martin-L{\”o}f randomness},
author={Yu, L.},
journal={Annals of Pure and Applied Logic},
year={2011},
publisher={Elsevier}
}