### Citations

397 |
The definition of random sequences
- Martin-Löf
- 1966
(Show Context)
Citation Context ... random sequence to be one that satisfies just those probabilistic laws contained in a chosen countable subfamily L ′ ⊆ L. The subfamily L ′ is typically specified using computabilitytheoretic (e.g., =-=[9]-=-) or logical (e.g., [3, 10]) definability criteria. Since there are many (reasonable) possibilities for this, a spectrum of different notions of randomness arises. The area is now a mature one, see [1... |

251 | Domain theory in logical form
- Abramsky
- 1991
(Show Context)
Citation Context ...e Axiom of Choice (AC), it is not possible to define a nontrivial measure on all subsets of Euclidean space R n that is invariant under the Euclidean group of isometries. By partitioning the interval =-=[0, 1]-=- into countably many (essentially) intertranslatable pieces, Vitali [24] showed that, in dimension 1, the existence of such a measure is ruled out by countable additivity. In dimension 3, the Banach-T... |

240 |
Algorithmic randomness and complexity
- Downey, Hirschfeldt
- 2010
(Show Context)
Citation Context ...9]) or logical (e.g., [3, 10]) definability criteria. Since there are many (reasonable) possibilities for this, a spectrum of different notions of randomness arises. The area is now a mature one, see =-=[11, 12]-=- for recent textbook surveys, and has important applications. Solovay’s notion of randomness, based on definability in a model of set theory, was used to prove the result, referred to above, that it i... |

160 |
A model of set-theory in which every set of reals is Lebesgue measurable
- Solovay
- 1970
(Show Context)
Citation Context ...measurable. The examples of Vitali and of Banach and Tarski make use of nonmeasurable sets, constructed using uncountable instances of AC. The application of AC in such examples is necessary. Solovay =-=[3]-=- showed that the measurability of every subset of R n is Preprint submitted to Elsevier November 22, 2010consistent if AC is weakened to Dependent Choice (DC). 1 Nevertheless, ZFC remains the preferr... |

156 |
An extension of the Galois theory of Grothendieck
- Joyal, Tierney
- 1984
(Show Context)
Citation Context ...ture work, we plan to give an alternative proof of Theorem 1 using Joyal and Tierney’s characterisation of the opposite of the lattice of sublocales via a universal property in the category of frames =-=[23]-=-; more accurately, using Madden’s variant of this result for σ-sublocales [18]. Using this approach, it should be possible to show that measures on O(X) extend to measures on S(X) for arbitrary σ-loca... |

154 | Computability and Randomness
- Nies
- 2009
(Show Context)
Citation Context ...9]) or logical (e.g., [3, 10]) definability criteria. Since there are many (reasonable) possibilities for this, a spectrum of different notions of randomness arises. The area is now a mature one, see =-=[11, 12]-=- for recent textbook surveys, and has important applications. Solovay’s notion of randomness, based on definability in a model of set theory, was used to prove the result, referred to above, that it i... |

127 |
Topology via Logic
- Vickers
- 1989
(Show Context)
Citation Context ...aper is to provide a resolution of the paradox that does lead to a canonical notion of randomness. We avoid the arbitrariness in the choice of L ′ , by retaining the idea that a random sequence 4 See =-=[5, 6, 7]-=- for further elaboration of this view. 5 This terminology is introduced for rhetorical effect in the present paper. Other “paradoxes of randomness” have been identified elsewhere, e.g., [8]. 3should ... |

62 |
Atomless parts of spaces
- Isbell
- 1972
(Show Context)
Citation Context ... notion of “part” is a natural and established one. We view spaces of interest (such as Rn ) as locales, and the notion of “part” is given by the standard notion of sublocale, introduced by Isbell in =-=[4]-=-. Locales (essentially) generalise topological spaces, by replacing the lattice of open sets with an abstract lattice whose elements are called opens. There is no requirement that opens be sets of poi... |

54 | Exploring Randomness
- Chaitin
- 2000
(Show Context)
Citation Context ...ee [1, 21, 23] for further elaboration of this view. 5 This terminology is introduced for rhetorical effect in the present paper. Other “paradoxes of randomness” have been identified elsewhere, e.g., =-=[6]-=-. 3the paradox that does lead to a canonical notion of randomness. We avoid the arbitrariness in the choice of L ′ , by retaining the idea that a random sequence should satisfy all measure 1 properti... |

45 |
Computability and randomness, volume 51 of Oxford Logic Guides
- Nies
- 2009
(Show Context)
Citation Context ...]) or logical (e.g., [22, 18]) definability criteria. Since there are many (reasonable) possibilities for this, a spectrum of different notions of randomness arises. The area is now a mature one, see =-=[19, 8]-=- for recent textbook surveys, and has important applications. Solovay’s notion of randomness, based on definability in a model of set theory, was used to prove the result, referred to above, that it i... |

44 |
Topology via Logic. Volume 5 of Cambridge Tracts
- Vickers
- 1989
(Show Context)
Citation Context ... provides a good abstraction for describing the fundamental stochastic notion of randomness from empirical experience. The second contribution of the present paper is to provide a resolution of 4 See =-=[1, 21, 23]-=- for further elaboration of this view. 5 This terminology is introduced for rhetorical effect in the present paper. Other “paradoxes of randomness” have been identified elsewhere, e.g., [6]. 3the par... |

36 |
Sur la décomposition des ensembles de points en parties respectivement congruentes, Fund
- Banach, Tarski
- 1924
(Show Context)
Citation Context ...y many (essentially) intertranslatable pieces, Vitali [24] showed that, in dimension 1, the existence of such a measure is ruled out by countable additivity. In dimension 3, the Banach-Tarski theorem =-=[4]-=- provides a striking demonstration of the impossibility of finite additivity: a solid sphere can be decomposed into finitely many disjoint pieces, which can then be reassembled to produce two solid sp... |

25 |
Valuations on continuous lattices
- Lawson
- 1982
(Show Context)
Citation Context ...y (12) i≥0 j≥0 7= sup sup µ(uj ∧ x ∧ yi) j≥0 i≥0 = sup i≥0 µ( ∨ (uj ∧ x ∧ yi)) by (5) j≥0 = sup µ(x ∧ yi) by (11) . i≥0 by (9) since µ( ∨ uj ∧ yi) < ∞ i≥0 In the literature on valuations, see, e.g., =-=[15, 2, 13]-=-, one often meets a stronger continuity requirement than σ-continuity. Definition 2.11. If L is a complete lattice with measure µ, we say that µ is continuous if, for every directed D ⊆ L, it holds th... |

20 | spaces, volume 3 of Cambridge Studies in Advanced Mathematics - Stone - 1986 |

19 |
On the notion of randomness
- Martin-Löf
- 1968
(Show Context)
Citation Context ...one that satisfies just those probabilistic laws contained in a chosen countable subfamily L ′ ⊆ L. The subfamily L ′ is typically specified using computabilitytheoretic (e.g., [9]) or logical (e.g., =-=[3, 10]-=-) definability criteria. Since there are many (reasonable) possibilities for this, a spectrum of different notions of randomness arises. The area is now a mature one, see [11, 12] for recent textbook ... |

17 |
problema della misura dei gruppi di punti di una retta, Bologna,Tip. Camberini e Parmeggiani
- Vital, Sul
- 1905
(Show Context)
Citation Context ...e Axiom of Choice (AC), it is not possible to define a nontrivial measure on all subsets of Euclidean space R n that is invariant under the Euclidean group of isometries. By partitioning the interval =-=[0, 1]-=- into countably many (essentially) intertranslatable pieces, Vitali [1] showed that, in dimension 1, the existence of such a measure is ruled out by countable additivity. In dimension 3, the Banach-Ta... |

7 |
Complete metric boolean algebra, Philosophical studies 77
- Kolmogorov
- 1995
(Show Context)
Citation Context ...siderable overlap in our approach with an old foundational tradition in measure and probability theory of basing measure/probability on algebras of parts/events quotiented modulo nullsets, see, e.g., =-=[21, 22]-=-. There are, however, two main novelties of the approach in this paper. The lesser novelty is that we focus on (σ-)frames rather than boolean algebras, which makes continuity the fundamental notion, a... |

6 | Extension of valuations on locally compact sober spaces. Topology ApPL
- Alvarez-Manilla
- 2002
(Show Context)
Citation Context ...y (12) i≥0 = sup sup µ(uj ∧ x ∧ yi) j≥0 i≥0 = sup i≥0 µ( ∨ (uj ∧ x ∧ yi)) by (5) j≥0 j≥0 = sup µ(x ∧ yi) by (11) . i≥0 by (9) since µ( ∨ uj ∧ yi) < ∞ i≥0 7In the literature on valuations, see, e.g., =-=[13, 14, 15]-=-, one often meets a stronger continuity requirement than σ-continuity. Definition 2.11. If L is a complete lattice with measure µ, we say that µ is continuous if, for every directed D ⊆ L, it holds th... |

5 |
Valuations on continuous lattices, Continuous Lattices and Related
- Lawson
- 1982
(Show Context)
Citation Context ...y (12) i≥0 = sup sup µ(uj ∧ x ∧ yi) j≥0 i≥0 = sup i≥0 µ( ∨ (uj ∧ x ∧ yi)) by (5) j≥0 j≥0 = sup µ(x ∧ yi) by (11) . i≥0 by (9) since µ( ∨ uj ∧ yi) < ∞ i≥0 7In the literature on valuations, see, e.g., =-=[13, 14, 15]-=-, one often meets a stronger continuity requirement than σ-continuity. Definition 2.11. If L is a complete lattice with measure µ, we say that µ is continuous if, for every directed D ⊆ L, it holds th... |

5 |
Quotient frames and subspaces
- Dowker, Papert
- 1966
(Show Context)
Citation Context ...es of X, partially ordered by ⊆. The next few results survey, without proof, various properties of σ-sublocales, all of which are standard. Corresponding results for sublocales can be found in, e.g., =-=[7, 10, 11]-=-. For the case of σ-(sub)locales, one can either adapt the proofs in [10] (which carry over), or refer to the more general treatment in [16], where σ-locales are subsumed as the κ = ℵ1 (sic) instance ... |

4 |
κ-frames
- Madden
- 1991
(Show Context)
Citation Context ...rresponding results for sublocales can be found in, e.g., [4, 17]. For the case of σ-(sub)locales, one can either adapt the proofs in [4] (which carry over), or refer to the more general treatment in =-=[18]-=-, where σ-locales are subsumed as the κ = ℵ1 instance of κ-locales (there treated purely algebraically as κ-frames). Proposition 3.7. For any σ-locale X, the partial order S(X) is a coframe, hence com... |

3 |
Measure extension theorems on T0-spaces. Topology and its Applications
- Keimel, Lawson
- 2005
(Show Context)
Citation Context ...y (12) i≥0 = sup sup µ(uj ∧ x ∧ yi) j≥0 i≥0 = sup i≥0 µ( ∨ (uj ∧ x ∧ yi)) by (5) j≥0 j≥0 = sup µ(x ∧ yi) by (11) . i≥0 by (9) since µ( ∨ uj ∧ yi) < ∞ i≥0 7In the literature on valuations, see, e.g., =-=[13, 14, 15]-=-, one often meets a stronger continuity requirement than σ-continuity. Definition 2.11. If L is a complete lattice with measure µ, we say that µ is continuous if, for every directed D ⊆ L, it holds th... |

2 |
Topology, in: Handbook of Logic in
- Smyth
- 1992
(Show Context)
Citation Context ...aper is to provide a resolution of the paradox that does lead to a canonical notion of randomness. We avoid the arbitrariness in the choice of L ′ , by retaining the idea that a random sequence 4 See =-=[5, 6, 7]-=- for further elaboration of this view. 5 This terminology is introduced for rhetorical effect in the present paper. Other “paradoxes of randomness” have been identified elsewhere, e.g., [8]. 3should ... |

2 |
Algebraic theory of measure and integration. Chelsea Publishing co., 2nd edition
- Carathéodory
- 1986
(Show Context)
Citation Context ...lap in the approach of this paper with an old foundational tradition in measure and probability theory of basing measure/probability on algebras of parts/events quotiented modulo nullsets, see, e.g., =-=[14, 5]-=-. There are, however, two main novelties in our approach. The lesser novelty is that we focus on (σ-)frames rather than boolean algebras, which makes continuity the fundamental notion, and allows the ... |

2 |
The locale of random sequences
- Simpson
- 2007
(Show Context)
Citation Context ...rmined as the intersection in S(2ω ) of all probabilistic laws. We briefly outline some of its other features. A more thorough treatment of these and other properties is planned for a follow-up paper =-=[20]-=-. Pointlessness. The first fact to observe about the locale Ran is that it has no points. The points of 2 ω are just the elements of 2 ω . For any such point α, the set 2 ω − {α} is open with measure ... |

1 |
Paradoxes of randomness, Complexity 7
- Chaitin
- 2002
(Show Context)
Citation Context ... See [5, 6, 7] for further elaboration of this view. 5 This terminology is introduced for rhetorical effect in the present paper. Other “paradoxes of randomness” have been identified elsewhere, e.g., =-=[8]-=-. 3should satisfy all measure 1 properties. No contradiction arises because we use this criterion to define a “random part” of 2ω that is nontrivial even though it (necessarily) contains no individua... |

1 |
A duality involving Borel spaces
- Baboolal, Ghosh
(Show Context)
Citation Context ...σ-algebra of measurable subsets of X. If Y is another measurable space then any measurable function f : X → Y determines a σ-continuous map of σ-locales. In the case that Y is “sober” in the sense of =-=[16]-=-, measurable functions from X to Y are in one-to-one corespondence with σcontinuous maps. More generally, any σ-complete-boolean algebra is a σ-locale, and any homomorphism of σ-complete-boolean algeb... |

1 |
The locale of random sequences, in preparation
- Halmos, Theory, et al.
- 1950
(Show Context)
Citation Context ...imply O(X) quotiented by the relation of null symmetric difference. The construction of the σ-frame O(Ran(µ)) is thus reminiscent of the construction of measure algebras in measure theory, see, e.g., =-=[19]-=-. This is discussed further in Examples 5.5 and 5.6 below. By Corollary 1, the measure µ ∗ restricts to give a measure µ↾ Ran(µ) on the σlocale Ran(µ). It follows easily that the σ-sublocale Ran(µ↾ Ra... |

1 |
Algebraic Theory of Measure and Integration, 2nd Edition, Chelsea Publishing Co
- Carathéodory
- 1988
(Show Context)
Citation Context ...siderable overlap in our approach with an old foundational tradition in measure and probability theory of basing measure/probability on algebras of parts/events quotiented modulo nullsets, see, e.g., =-=[21, 22]-=-. There are, however, two main novelties of the approach in this paper. The lesser novelty is that we focus on (σ-)frames rather than boolean algebras, which makes continuity the fundamental notion, a... |

1 |
Complete metric boolean algebras
- Kolmogorov
- 1995
(Show Context)
Citation Context ...lap in the approach of this paper with an old foundational tradition in measure and probability theory of basing measure/probability on algebras of parts/events quotiented modulo nullsets, see, e.g., =-=[14, 5]-=-. There are, however, two main novelties in our approach. The lesser novelty is that we focus on (σ-)frames rather than boolean algebras, which makes continuity the fundamental notion, and allows the ... |