A Parametrization Theorem for Generalized Borel Sets

AbstractThis paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

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Convex cores of measures on R d

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.38.2001.1-4.12 ◽

2001 ◽

Vol 38 (1-4) ◽

pp. 177-190 ◽

Cited By ~ 1

Author(s):

Imre Csiszár ◽

F. Matúš

Keyword(s):

Borel Measure ◽

Convex Sets ◽

Probability Measures ◽

Countable Number ◽

Borel Sets ◽

Finite Borel Measure ◽

Number Of Faces ◽

Convex Core

We define the convex core of a finite Borel measure Q on R d as the intersection of all convex Borel sets C with Q(C) =Q(R d). It consists exactly of means of probability measures dominated by Q. Geometric and measure-theoretic properties of convex cores are studied, including behaviour under certain operations on measures. Convex cores are characterized as those convex sets that have at most countable number of faces.

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Γ‎-Null and Γ‎N-Null Sets

Frechet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179) ◽

10.23943/princeton/9780691153551.003.0005 ◽

2012 ◽

Author(s):

Joram Lindenstrauss ◽

David Preiss ◽

Tiˇser Jaroslav

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Finite Dimension ◽

Baire Category ◽

Borel Sets ◽

Basic Properties ◽

Gâteaux Differentiability ◽

Gateaux Differentiability ◽

Null Set

This chapter introduces the notions of Γ‎-null and Γ‎ₙ-null sets, which are σ‎-ideals of subsets of a Banach space X. Γ‎-null set is key for the strongest known general Fréchet differentiability results in Banach spaces, whereas Γ‎ₙ-null set presents a new, more refined concept. The reason for these notions comes from an (imprecise) observation that differentiability problems are governed by measure in finite dimension, but by Baire category when it comes to behavior at infinity. The chapter first relates Γ‎-null and Γ‎ₙ-null sets to Gâteaux differentiability before discussing their basic properties. It then describes Γ‎-null and Γ‎ₙ-null sets of low Borel classes and presents equivalent definitions of Γ‎ₙ-null sets. Finally, it considers the separable determination of Γ‎-nullness for Borel sets.

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