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Measure and Dimension Functions: Measurability and Densities
 Mathematical Proceedings of the Cambridge Philosophical Society
, 1997
"... this paper, we find a basic limitation on this possibility. We do this by determining the descriptive set theoretic complexity of the packing functions. Whereas the Hausdorff dimension function on the space of compact sets is Borel measurable, the packing dimension function is not. On the other hand ..."
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Cited by 13 (2 self)
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this paper, we find a basic limitation on this possibility. We do this by determining the descriptive set theoretic complexity of the packing functions. Whereas the Hausdorff dimension function on the space of compact sets is Borel measurable, the packing dimension function is not. On the other hand, we show that the packing measure and dimension functions are measurable with respect to the oealgebra generated by the analytic sets. Thus, the usual sorts of measurability properties used in connection with Hausdorff measure,
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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Cited by 8 (0 self)
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Trichotomies for ideals of compact sets
 J. SYMBOLIC LOGIC
"... We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal. ..."
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Cited by 3 (1 self)
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We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal.
Reducing subspaces
, 1993
"... Let T be a selfadjoint operator acting in a separable Hilbert space H. We establish a correspondence between the reducing subspaces of T that come from a spectral projection and the convex, normclosed bands in the set of finite Borel measures on R. If H is not separable, we still obtain a reducing ..."
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Cited by 2 (1 self)
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Let T be a selfadjoint operator acting in a separable Hilbert space H. We establish a correspondence between the reducing subspaces of T that come from a spectral projection and the convex, normclosed bands in the set of finite Borel measures on R. If H is not separable, we still obtain a reducing subspace corresponding to each convex normclosed band. These observations lead to a unified treatment of various reducing subspaces; moreover, they also settle some open questions and suggest new decompositions. 1 Reducing subspaces and bands Throughout this paper, we fix a selfadjoint operator T acting in Hilbert space H. As T is selfadjoint, it admits the representation T = � λ dE(λ) where E(·) R
The Structure Of The sigmaIdeal Of sigmaPorous Sets
, 1999
"... . We show a general method of construction of nonoeporous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each nonoeporous Suslin subset of a topologically complete metric space contains a nonoeporous closed subset. We show also a suff ..."
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. We show a general method of construction of nonoeporous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each nonoeporous Suslin subset of a topologically complete metric space contains a nonoeporous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a nonoeporous element. Namely, if we denote the space of all compact subsets of a compact metric space E with the Hausdorff metric by K(E), then it is shown that each analytic subset of K(E) containing all countable compact subsets of E contains necessarily an element, which is nonoeporous subset of E. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed nonoeporous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the oeideal of compact oeporous sets. 1. Introduction Let (P; ae) be a...
ON SOME ERRORS RELATED TO THE GRADUATION OF MEASURING INSTRUMENTS
, 2006
"... Abstract. The error on a real quantity Y due to the graduation of the measuring instrument may be approximately represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator does not depend on the probability law of Y as soon as this law possesses a ..."
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Abstract. The error on a real quantity Y due to the graduation of the measuring instrument may be approximately represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator does not depend on the probability law of Y as soon as this law possesses a continuous density. This feature is related to the “arbitrary functions principle ” (Poincaré, Hopf). We give extensions of this property to R d and to the Wiener space for some approximations of the Brownian motion. We use a Girsanov theorem for Dirichlet forms which has its own interest. Connections are given with the discretization of stochastic differential equations.
unknown title
, 2007
"... Analytic representation of functions and a new quasianalyticity threshold By Gady Kozma and Alexander Olevskiĭ* We characterize precisely the possible rate of decay of the antianalytic half of a trigonometric series converging to zero almost everywhere. 1. ..."
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Analytic representation of functions and a new quasianalyticity threshold By Gady Kozma and Alexander Olevskiĭ* We characterize precisely the possible rate of decay of the antianalytic half of a trigonometric series converging to zero almost everywhere. 1.
THE LIMITS OF DETERMINACY IN SECOND ORDER ARITHMETIC
, 2010
"... We establish the precise bounds for the amount of determinacy provable in second order arithmetic. We show that for every natural number n, second order arithmetic can prove that determinacy holds for Boolean combinations of n many Π 0 3 classes, but it cannot prove that all finite Boolean combina ..."
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We establish the precise bounds for the amount of determinacy provable in second order arithmetic. We show that for every natural number n, second order arithmetic can prove that determinacy holds for Boolean combinations of n many Π 0 3 classes, but it cannot prove that all finite Boolean combinations of Π 0 3 classes are determined. More specifically, we prove that Π 1 n+2CA 0 ⊢ nΠ 0 3DET, but that ∆ 1 n+2CA � nΠ 0 3DET, where nΠ 0 3 is the nth level in the difference hierarchy of Π 0 3 classes. We also show some conservativity results that imply that reversals for the theorems above are not possible. We prove that for every true Σ 1 4 sentence T (as for instance nΠ 0 3DET) and