## Effectively closed sets of measures and randomness

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Venue: | Ann. Pure Appl. Logic |

Citations: | 7 - 1 self |

### BibTeX

@ARTICLE{Reimann_effectivelyclosed,

author = {Jan Reimann},

title = {Effectively closed sets of measures and randomness},

journal = {Ann. Pure Appl. Logic},

year = {},

pages = {170--182}

}

### OpenURL

### Abstract

We show that if a real x ∈ 2ω is strongly Hausdorff Hh-random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure µ such that the µ-measure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π0 1-classes applied to closed sets of probability measures. We use the main result to give a new proof of Frostman’s Lemma, to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman’s Theorem. 1

### Citations

1690 | Kolmogorov complexity and its applications
- Li, Vitanyi
- 1990
(Show Context)
Citation Context ...th the basic notions of measure theory and descriptive set theory, as treated in [15]. Furthermore, we presuppose some knowledge in algorithmic randomness and computability theory, as can be found in =-=[26]-=- or [5]. 2 Measures and Randomness on Cantor Space In this section we briefly review the basic notions of measure on the Cantor space 2 ω . We make use of the special topological structure of 2 ω to g... |

540 |
Fractal Geometry: Mathematical Foundations and Applications
- Falconer
- 2003
(Show Context)
Citation Context ...s (A) = 0}. Hausdorff dimension is a central notion in fractal geometry and has recently received a lot of attention in the effective setting, too. We will not dwell further on this here but refer to =-=[7, 28, 27, 34]-=-. 2.4 Representations of premeasures To define randomness, we want to incorporate measures into the effective aspects of a randomness test. For this purpose, we have to represent it in a form that mak... |

408 |
Geometry of sets and measures in Euclidean spaces
- Mattila
- 1995
(Show Context)
Citation Context ...ity is fractal geometry. While it initially studied fractal properties of sets in Euclidean or other metric spaces, the geometric analysis is now widely applied to measures, too. The books by Mattila =-=[28]-=- and Edgar [6] reflect this quite well. A cornerstone of this development was the work by Frostman [8]. He realized that there is a close connection between the Hausdorff dimension of a set and the en... |

326 |
Classical descriptive set theory
- KECHRIS
- 1995
(Show Context)
Citation Context ...or packing dimension, and discuss an open question as well as directions for further research. We assume familiarity with the basic notions of measure theory and descriptive set theory, as treated in =-=[15]-=-. Furthermore, we presuppose some knowledge in algorithmic randomness and computability theory, as can be found in [26] or [5]. 2 Measures and Randomness on Cantor Space In this section we briefly rev... |

195 |
Descriptive set theory
- Moschovakis
- 1980
(Show Context)
Citation Context ...herefore, in the following we give a succinct description of how to devise a Π0 1-class in 2ω of representations of probability measures. In general, our approach follows the framework of Moschovakis =-=[32]-=-, with a few adaptations regarding compactness. Each step can be justified by resorting to the accordant results in [32]. 2.5 The space of probability measures on Cantor space Recall that P denotes th... |

161 |
Algorithmic Randomness and Complexity
- Downey, Hirschfeldt
- 2010
(Show Context)
Citation Context ...asic notions of measure theory and descriptive set theory, as treated in [15]. Furthermore, we presuppose some knowledge in algorithmic randomness and computability theory, as can be found in [26] or =-=[5]-=-. 2 Measures and Randomness on Cantor Space In this section we briefly review the basic notions of measure on the Cantor space 2 ω . We make use of the special topological structure of 2 ω to give a u... |

107 |
On the notion of a random sequence
- Levin
- 1973
(Show Context)
Citation Context ... representation of µ such that the real is µ-random relative to that representation. 3 Effectively closed sets of measures, randomness, and capacities 3.1 Effectively closed sets and randomness Levin =-=[23]-=- was the first to use Π0 1 classes of measures in algorithmic randomness. He observed that, given a test W, the set of probability measures µ such that W is correct for µ is Π0 1 . Levin was intereste... |

104 |
Hausdorff measures
- Rogers
- 1970
(Show Context)
Citation Context ... if and only if σ ⊂ x. Finally, given U ⊆ 2 <ω , we write NU to denote the open set induced by U, i.e. NU = ⋃ σ∈U Nσ. The following method to construct outer measures has been referred to as Method I =-=[33, 39]-=-. Definition 1. Let 2 <ω be the set of all finite binary sequences. A premeasure is a mapping ρ : 2 <ω → R ≥0 . 3If ρ is a premeasure, define the set function µρ : P(2 ω ) → R ≥0 by letting ⎧ ⎫ ⎪⎨ ∑ ... |

102 |
Randomness conservation inequalities� information and independence
- Levin
- 1984
(Show Context)
Citation Context ...lative to z and representation pρ, if it passes all pρ ⊕ z-tests which are correct for µρ. An obvious objection to this definition of randomness is that it is representation dependent. In fact, Levin =-=[24, 25]-=- and recently Gács [10] have given definitions of randomness independent of the underlying measure. There are arguments in favor of and against both approaches (see also [35]). In the context of this ... |

95 | The dimensions of individual strings and sequences
- Lutz
(Show Context)
Citation Context ...s (A) = 0}. Hausdorff dimension is a central notion in fractal geometry and has recently received a lot of attention in the effective setting, too. We will not dwell further on this here but refer to =-=[7, 28, 27, 34]-=-. 2.4 Representations of premeasures To define randomness, we want to incorporate measures into the effective aspects of a randomness test. For this purpose, we have to represent it in a form that mak... |

48 |
On partial randomness
- Calude, Staiger, et al.
- 2006
(Show Context)
Citation Context ...ng premeasure, i.e. we will write µ(σ) instead of µ(Nσ). If ρ is a probability premeasure, then µρ is a Borel measure, i.e. all Borel sets are measurable, and their measure is a finite real number in =-=[0, 1]-=-. It will later be important to identify the subset of 2 ω on which a probability measure ‘resides’. The support supp(µ) of a probability measure µ is the smallest closed set F ⊆ 2 ω such that µ(2 ω \... |

47 |
Π 0 1 classes, and complete extensions of PA
- Measure
- 1984
(Show Context)
Citation Context ...) h-bounded. Theorem 5 (Effective Capacitability Theorem). Suppose x ∈ 2 ω is strongly hcomplex, where h is a computable, convex order function. Then x is h-capacitable. Proof. By a theorem of Kučera =-=[22]-=- and Gács [9] there exists a Martin-Löf random real R such that x ≤T R. In fact, there exists a Turing functional Φ and a Π 0 1 set B ⊆ 2 ω such that all elements of B are Martin-Löf random and for an... |

46 | Kolmogorov complexity and the recursion theorem
- Kjos-Hanssen, Merkle, et al.
(Show Context)
Citation Context ...ective capacitability, hence the following result completely describes to what extend Frostman’s Lemma holds effectively. 12First, we introduce strong h-complexity. Kjos-Hanssen, Merkle, and Stephan =-=[21]-=- defined a real to be complex if there exists a computable order function h such that (∀n) [K(x↾n) ≥ h(n)], (3.5) where K denotes prefix-free Kolmogorov complexity. If x is complex via h, then we call... |

42 |
Every sequence is reducible to a random one
- Gacs
- 1986
(Show Context)
Citation Context ...heorem 5 (Effective Capacitability Theorem). Suppose x ∈ 2 ω is strongly hcomplex, where h is a computable, convex order function. Then x is h-capacitable. Proof. By a theorem of Kučera [22] and Gács =-=[9]-=- there exists a Martin-Löf random real R such that x ≤T R. In fact, there exists a Turing functional Φ and a Π 0 1 set B ⊆ 2 ω such that all elements of B are Martin-Löf random and for any y ∈ 2 ω the... |

36 | Uniform test of algorithmic randomness over a general space
- Gács
(Show Context)
Citation Context ...structure later, in the following we introduce an alternative representation for probability measures. An effective representation of probability measures has been developed elsewhere (for example in =-=[10]-=-), but we found none of the accounts completely adequate for our purposes. Therefore, in the following we give a succinct description of how to devise a Π0 1-class in 2ω of representations of probabil... |

31 | Brownian Motion
- Mörters, Peres
- 2010
(Show Context)
Citation Context ...ence of measures such that any limit point in the weak topology (which exists by compactness) will have the desired properties (see [28]). Alternatively, one can use the MaxFlow-MinCut Theorem (as in =-=[31]-=-). A different approach, which works in general metric spaces, was given by Howroyd [13], introducing weighted Hausdorff measures and using the Hahn-Banach Theorem. These proofs make essential use of ... |

24 |
Integral, probability, and fractal measures
- Edgar
- 1998
(Show Context)
Citation Context ...geometry. While it initially studied fractal properties of sets in Euclidean or other metric spaces, the geometric analysis is now widely applied to measures, too. The books by Mattila [28] and Edgar =-=[6]-=- reflect this quite well. A cornerstone of this development was the work by Frostman [8]. He realized that there is a close connection between the Hausdorff dimension of a set and the energies of meas... |

22 |
Extracting information is
- Miller
- 2008
(Show Context)
Citation Context ... −n 2 −k(t−s) . Hence ∑ {2 −|σ|t : σ ∈ Wn} ≤ 2 −n ∑ k 2 −k(t−s) = 2 −n 1 . 1 − 2−(t−s) □ This answers a question by Kjos-Hanssen. The result was independently obtained by Kjos-Hanssen [16] and Miller =-=[30]-=-. Finally, we use the effective capacitability theorem to give a new characterization of effective dimension, which reflects the duality between the complexity of a real and the capacity of a measure ... |

21 | Relativizing Chaitin’s halting probability
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...t U such that for any x that passes U there exists a measure µ ∈ S such that x is µ-random. A result in a similar spirit has recently been shown independently by Downey, Hirschfeldt, Miller, and Nies =-=[4]-=- and Reimann and Slaman [37]. 10Theorem 2. Let z ∈ 2 ω , and let T ⊆ 2 <ω be an infinite tree recursive in z. Then, for every real R which is λ-random relative to z, there is an infinite path y throu... |

21 |
Uniform tests of randomness
- Levin
- 1976
(Show Context)
Citation Context ...lative to z and representation pρ, if it passes all pρ ⊕ z-tests which are correct for µρ. An obvious objection to this definition of randomness is that it is representation dependent. In fact, Levin =-=[24, 25]-=- and recently Gács [10] have given definitions of randomness independent of the underlying measure. There are arguments in favor of and against both approaches (see also [35]). In the context of this ... |

20 |
Introduction to Measure and Integration
- Munroe
- 1953
(Show Context)
Citation Context ... if and only if σ ⊂ x. Finally, given U ⊆ 2 <ω , we write NU to denote the open set induced by U, i.e. NU = ⋃ σ∈U Nσ. The following method to construct outer measures has been referred to as Method I =-=[33, 39]-=-. Definition 1. Let 2 <ω be the set of all finite binary sequences. A premeasure is a mapping ρ : 2 <ω → R ≥0 . 3If ρ is a premeasure, define the set function µρ : P(2 ω ) → R ≥0 by letting ⎧ ⎫ ⎪⎨ ∑ ... |

17 |
The complexity of finite objects and the basing of the concepts of information and randomness on the theory of algorithms
- Zvonkin, Levin
- 1970
(Show Context)
Citation Context ...omplexity for reals by replacing K in (3.5) by another type of Kolmogorov complexity. A (continuous) semimeasure is a function η : 2 <ω → [0, 1] such that ∀σ [η(σ) ≥ η(σ ⌣ 0) + η(σ ⌣ 1)]. (3.6) Levin =-=[41]-=- proved the existence of an optimal enumerable semimeasure M. A semimeasure is enumerable if the set {(σ, q) ∈ 2 <ω × Q: q < η(σ)} is r.e. For any enumerable semimeasure η there exists a constant c su... |

13 | Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions - Frostman - 1935 |

11 |
Computability and fractal dimension. Doctoral dissertation, Universität
- Reimann
- 2004
(Show Context)
Citation Context ...s (A) = 0}. Hausdorff dimension is a central notion in fractal geometry and has recently received a lot of attention in the effective setting, too. We will not dwell further on this here but refer to =-=[7, 28, 27, 34]-=-. 2.4 Representations of premeasures To define randomness, we want to incorporate measures into the effective aspects of a randomness test. For this purpose, we have to represent it in a form that mak... |

11 | Measures and their random reals
- Reimann, Slaman
(Show Context)
Citation Context ...at the study of duality between the complexity of sets and measures leads to interesting insights and questions when transferred to an effective setting. Recent work by Theodore Slaman and the author =-=[37, 36]-=- gives evidence for this. While a real is non-trivially random (i.e. there exists a probability measure such that x is µ-random and µ({x}) = 0, see [37]) if and only if it is not recursive, the questi... |

8 |
Members of random closed sets
- Diamondstone, Kjos-Hanssen
- 2009
(Show Context)
Citation Context ...This raises the hope that the effective theory might in turn contribute to the classical setting, via relativization, as it did in the case of effective descriptive set theory. Recently, Kjos-Hanssen =-=[18, 19]-=- has obtained results in this direction. The outline of the paper is as follows. In Section 2, we first present a brief account of measure theory on Cantor space. The basic notion is that of an outer ... |

7 |
Degrees of Random Sequences
- Kautz
- 1991
(Show Context)
Citation Context ...resting to have a classification of the reals which do compute a λ-random real. One way to ensure this is to be random for a measure which the real itself computes (by results of Levin [41] and Kautz =-=[14]-=-). Question. For which strongly H h -random x does there exist a continuous measure µ ≤T x such that x is µ-random? It follows directly from the construction in the proof of Theorem 5 that the complex... |

7 |
Randomness for continuous measures
- Reimann, Slaman
(Show Context)
Citation Context ...at the study of duality between the complexity of sets and measures leads to interesting insights and questions when transferred to an effective setting. Recent work by Theodore Slaman and the author =-=[37, 36]-=- gives evidence for this. While a real is non-trivially random (i.e. there exists a probability measure such that x is µ-random and µ({x}) = 0, see [37]) if and only if it is not recursive, the questi... |

6 |
Strong and weak duality principles for fractal dimension in Euclidean space
- Cutler
- 1995
(Show Context)
Citation Context ...ul to a similar extent when dealing with randomness for Hausdorff measures, since probability measures are usually nicer to deal with. Frostman’s Lemma has been extended to packing measures by Cutler =-=[3]-=-. For details on packing measures and packing dimension, see [7]. Theorem 10. If A is a compact subset of 2 ω with dimP(A) ≥ t > 0, then there exists a probability measure µ such that supp(µ) ⊆ A, and... |

6 | Diagonally non-recursive functions and effective Hausdorff dimension
- Greenberg, Miller
(Show Context)
Citation Context ...ial effective Hausdorff dimension can, although the computational properties of such reals can be quite different from Martin-Löf random reals (see recent work by Miller [30] and Greenberg and Miller =-=[11]-=-). Open questions Recent breakthroughs by Miller [30] and Greenberg and Miller [11] on the socalled dimension problem have shown that not every H h -random real computes a λ-random real, even when h i... |

5 | Effective Hausdorff dimension
- Reimann, Stephan
- 2005
(Show Context)
Citation Context ...of notation, also be denoted by Hh . Finally an order is called convex, if for all n, h(n + 1) ≤ h(n) + 1. Let H be the set of premeasures corresponding to convex order functions. Reimann and Stephan =-=[38]-=- studied the class of geometrical premeasures. These satisfy the following condition: There are real numbers p, q such that 5(G1) 1/2 ≤ p < 1 and 1 ≤ q < 2; (G2) (∀σ ∈ 2 <ω ) (∀i ∈ {0, 1}) [ρ(σ ⌣ i) ... |

3 |
Computing Brownian slow points
- Kjos-Hanssen
(Show Context)
Citation Context ...This raises the hope that the effective theory might in turn contribute to the classical setting, via relativization, as it did in the case of effective descriptive set theory. Recently, Kjos-Hanssen =-=[18, 19]-=- has obtained results in this direction. The outline of the paper is as follows. In Section 2, we first present a brief account of measure theory on Cantor space. The basic notion is that of an outer ... |

2 | Effective packing dimension of Π0 1-classes
- Conidis
(Show Context)
Citation Context ...uch that supp(µ) ⊆ A, and for each x ∈ 2 ω it holds for infinitely many n that µ(x↾n) ≤ γ2 −nt . 22The effective analogue of this does not hold, in fact it fails in a striking way. Recently, Conidis =-=[2]-=- has constructed a countable Π0 1-class of effective packing dimension 1. By an observation of Kjos-Hanssen and Montalbán [20], no member of a countable Π0 1-class can be random for a continuous measu... |

2 |
Draft of a paper on Chaitin’s work
- Solovay
- 1975
(Show Context)
Citation Context ...nt, since any ρ-test is also an η-test for any η ≤ ρ with ρ ≤T η. (a) Martin-Löf randomness. This is given by RML(pρ, k) ≡ Premeasure(ρ) & (∀n) [∑ {ρ(σ): σ ∈ W ρ k,n } ≤ 2−n] . (b) Solovay randomness =-=[40]-=-. [ RS (pρ, k) ≡ Premeasure(ρ) & (∀n) Wk,n+1 ⊆ Wk,n & ∑ |Wk,n \ Wk,n+1| � 0 and finite & {ρ(σ): σ ∈ W ρ ] k,1 } ≤ 1 . (c) Strong randomness (Calude, Staiger, and Terwijn [1]). Here we need to quantify... |

1 |
Fractal geometry in complexity classes. SIGACT News Complexity Theory Column
- Hitchcock, Lutz, et al.
- 2005
(Show Context)
Citation Context ...apacities and effective dimension Given a randomness notion R, one can define a corresponding effective Hausdorff dimension by putting dim R H (x) = inf{s ∈ Q: x is not Hs -random for R}. We refer to =-=[7, 28, 27, 34, 12]-=- for definitions and background on classical and effective dimension concepts. Although it has been shown in [38] that various randomness notions do not agree on Hausdorff measures and yield a strict ... |

1 |
Infinite subsets of random sets
- Kjos-Hanssen
(Show Context)
Citation Context ... ∈ W (k) n } ≤ 2 −n 2 −k(t−s) . Hence ∑ {2 −|σ|t : σ ∈ Wn} ≤ 2 −n ∑ k 2 −k(t−s) = 2 −n 1 . 1 − 2−(t−s) □ This answers a question by Kjos-Hanssen. The result was independently obtained by Kjos-Hanssen =-=[16]-=- and Miller [30]. Finally, we use the effective capacitability theorem to give a new characterization of effective dimension, which reflects the duality between the complexity of a real and the capaci... |

1 |
Email correspondence
- Kjos-Hanssen
- 2008
(Show Context)
Citation Context ...overs the same reals that W does. □ Finally we note that a similar argument yields that strong randomness is a necessary condition for effective capacitability. This was observed by Bjørn KjosHanssen =-=[17]-=-. 20Corollary (Kjos-Hanssen). If x is not strongly H h -random then x is not effectively h-capacitable. Proof. Assume x is not strongly H h -random. Let µ be an h-bounded probability measure. Let W =... |

1 | Randomness - beyond Lebesgue measure
- Reimann
(Show Context)
Citation Context ...e call the representation given by P the Cauchy representation of P. 2.6 Randomness We briefly review the definition of randomness in the sense of Martin-Löf for arbitrary outer measures. We refer to =-=[35, 37]-=- for more details on this approach to randomness for arbitrary measures. Martin-Löf’s concept of randomness is based on the fact that every nullset for a measure defined via Method I (and Method II, a... |

1 |
Effective packing dimension ofΠ 0 1-classes
- Conidis
(Show Context)
Citation Context ...sureµsuch that supp(µ)⊆A, and for each x∈2 ω it holds for infinitely many n that µ(x↾n)≤γ2 −nt . 22The effective analogue of this does not hold, in fact it fails in a striking way. Recently, Conidis =-=[2]-=- has constructed a countableΠ 0 1-class of effective packing dimension 1. By an observation of Kjos-Hanssen and Montalbán [20], no member of a countableΠ 0 1-class can be random for a continuous measu... |