## Almost everywhere domination

Venue: | J. Symbolic Logic |

Citations: | 33 - 16 self |

### BibTeX

@ARTICLE{Dobrinen_almosteverywhere,

author = {Natasha L. Dobrinen and Stephen G. Simpson},

title = {Almost everywhere domination},

journal = {J. Symbolic Logic},

year = {},

volume = {69},

pages = {914--922}

}

### OpenURL

### Abstract

ATuringdegreea is said to be almost everywhere dominating if, for almost all X ∈ 2 ω with respect to the “fair coin ” probability measure on 2 ω,andforallg: ω → ω Turing reducible to X, thereexistsf: ω → ω of Turing degree a which dominates g. We study the problem of characterizing the almost everywhere dominating Turing degrees and other, similarly defined classes of Turing degrees. We relate this problem to some questions in the reverse mathematics of measure theory. 1

### Citations

476 | Recursively Enumerable Sets and Degrees - Soare - 1987 |

197 |
Subsystems of Second Order Arithmetic
- Simpson
- 1999
(Show Context)
Citation Context ...ining the weakest set existence axioms needed to prove specific mathematical theorems. This is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson =-=[12]-=-. Other results on the reverse mathematics of 6smeasure theory are in the papers of Yu [14, 15, 16, 17, 18], Yu/Simpson [19], and Brown/Giusto/Simpson [1]. A well known result in measure theory assert... |

63 | Recursion-Theoretic Hierarchies - Hinman - 1978 |

53 |
Classes of recursively enumerable sets and degrees of unsolvability
- Martin
- 1966
(Show Context)
Citation Context ...ee. The following are pairwise equivalent. 1. a is almost everywhere dominating. 2. a is uniformly almost everywhere dominating. 3. a ′ ≥ 0 ′′ . Toward Conjecture 2.9, the following theorem of Martin =-=[6]-=- is well known. Say that A is uniformly dominating if there exists f ∈ REC[A] such that f dominates every g ∈ REC. Again, this is a property of the Turing degree of A. Theorem 2.10 (Martin [6]). ATuri... |

51 |
Randomness and Genericity in the Degrees of Unsolvability
- Kurtz
- 1981
(Show Context)
Citation Context ...ecursive functions, is not almost everywhere dominating. In this paper we raise the problem of characterizing the Turing degrees which are almost everywhere dominating. The following theorem of Kurtz =-=[5]-=- implies that 0 ′ , the Turing degree of the Halting Problem, is almost everywhere dominating. We consider an apparently more restrictive property. Definition 2.2. We say that A ∈ 2ω is almost everywh... |

46 | Degrees of Random Sets - Kautz - 1991 |

8 | Π 0 1 sets and models of WKL0
- Simpson
- 2000
(Show Context)
Citation Context ...d subset of 2ω , hence for each n, � � � X {e} (n) � X ∈ Pe,i is finite, by compactness of 2 ω .Thuswehave ∀e ∀i ∀n ∃m ∀X (P(〈0,...,0, 1〉 � �� � e � X,i) ⇒{e} X (n) ≤ m). Now, by Lemma 3.5 of Simpson =-=[11]-=- relativized to A, the predicate ⎛ ∀X ⎝P(〈0,...,0, 1〉 � �� � e � X,i) ⇒{e} X ⎞ (n) ≤ m⎠ is Σ 0,A 1 . Hence by Σ 0,A 1 uniformization we find g : ω × ω × ω → ω recursive in A such that ∀e ∀X ∀i (P(〈0,.... |

5 |
Measure-theoretic uniformity in recursion theory and set theory
- Sacks
- 1969
(Show Context)
Citation Context ... ordinals belonging to M. It is known that, for almost all X ∈ 2ω , M[X] isamodelofZFC. This leads to a forcing-free proof of the independence of the Continuum Hypothesis. See the exposition of Sacks =-=[8]-=-. The purpose of this paper is to investigate recursion-theoretic analogs of Theorem 1.1, replacing the set-theoretic ground model M by the recursiontheoretic ground model REC = {f ∈ ωω | f is recursi... |

5 |
Measure Theory in Weak Subsystems of Second Order Arithmetic
- Yu
- 1987
(Show Context)
Citation Context ... is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of 6smeasure theory are in the papers of Yu =-=[14, 15, 16, 17, 18]-=-, Yu/Simpson [19], and Brown/Giusto/Simpson [1]. A well known result in measure theory asserts that the fair coin measure µ is regular. This means that measurable sets are approximable from within by ... |

5 |
Lebesgue convergence theorems and reverse mathematics
- Yu
- 1994
(Show Context)
Citation Context ... is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of 6smeasure theory are in the papers of Yu =-=[14, 15, 16, 17, 18]-=-, Yu/Simpson [19], and Brown/Giusto/Simpson [1]. A well known result in measure theory asserts that the fair coin measure µ is regular. This means that measurable sets are approximable from within by ... |

4 |
Riesz representation theorem, Borel measures and subsystems of second-order arithmetic
- Yu
- 1993
(Show Context)
Citation Context ... is carried out in the context of subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of 6smeasure theory are in the papers of Yu =-=[14, 15, 16, 17, 18]-=-, Yu/Simpson [19], and Brown/Giusto/Simpson [1]. A well known result in measure theory asserts that the fair coin measure µ is regular. This means that measurable sets are approximable from within by ... |

4 |
Measure theory and weak König’s lemma
- Yu, Simpson
- 1990
(Show Context)
Citation Context ... subsystems of second order arithmetic. For general background, see Simpson [12]. Other results on the reverse mathematics of 6smeasure theory are in the papers of Yu [14, 15, 16, 17, 18], Yu/Simpson =-=[19]-=-, and Brown/Giusto/Simpson [1]. A well known result in measure theory asserts that the fair coin measure µ is regular. This means that measurable sets are approximable from within by Fσ sets and from ... |

3 | Logic and Computation. Contemporary Mathematics - Sieg, editor - 1990 |

3 |
Radon-Nikodym theorem is equivalent to arithmetical comprehension
- Yu
- 1990
(Show Context)
Citation Context |

2 |
A study of singular points and supports of measures in reverse mathematics
- Yu
- 1996
(Show Context)
Citation Context |

2 |
Degrees of Random Sets. PhD thesis, Cornell University,1991. X + 89 pages. [18] Bjo/rn Kjos-Hanssen. Low for random reals and positive-measure domina-tion
- Kautz
- 2005
(Show Context)
Citation Context ....18] (a consequence of the Low Basis Theorem) that there exists an !-model of WKL0 in which (?) fails. We thank the refereefor suggesting this observation. Furthermore, using Theorem III.2.1 of Kautz =-=[4]-=-, we can build an !-model M of WKL0 such that (8X2M )(9Y 2M )(Y is !-random relative to X), and (8X2M )(9Y 2M )(Y is !-generic relative to X), yet(8 Y 2M )(Y 0 6>=T 000), hence (?) fails in M . Refere... |

1 |
Measure, category, and degrees of unsolvability
- Martin
- 1967
(Show Context)
Citation Context ....1, it would be natural to conjecture that for almost all X ∈ 2ω and all g ∈ REC[X] thereexistsf∈REC such that f dominates g. However, this is not the case, as shown by the following result of Martin =-=[7]-=-. Since the proof of Theorem 1.2 has not been published, we present it below. Theorem 1.2 (Martin [7]). For almost all X ∈ 2ω there exists g ∈ REC[X] such that g is not dominated by any f ∈ REC. Proof... |