Displaying similar documents to “Strong measure zero and meager-additive sets through the prism of fractal measures”

On Meager Additive and Null Additive Sets in the Cantor Space 2 ω and in ℝ

Tomasz Weiss (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let T be the standard Cantor-Lebesgue function that maps the Cantor space 2 ω onto the unit interval ⟨0,1⟩. We prove within ZFC that for every X 2 ω , X is meager additive in 2 ω iff T(X) is meager additive in ⟨0,1⟩. As a consequence, we deduce that the cartesian product of meager additive sets in ℝ remains meager additive in ℝ × ℝ. In this note, we also study the relationship between null additive sets in 2 ω and ℝ.

A note on the super-additive and sub-additive transformations of aggregation functions: The multi-dimensional case

Fateme Kouchakinejad, Alexandra Šipošová (2017)

Kybernetika

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For an aggregation function A we know that it is bounded by A * and A * which are its super-additive and sub-additive transformations, respectively. Also, it is known that if A * is directionally convex, then A = A * and A * is linear; similarly, if A * is directionally concave, then A = A * and A * is linear. We generalize these results replacing the directional convexity and concavity conditions by the weaker assumptions of overrunning a super-additive function and underrunning a sub-additive function, respectively. ...

More remarks on the intersection ideal 𝒩

Tomasz Weiss (2018)

Commentationes Mathematicae Universitatis Carolinae

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We prove in ZFC that every 𝒩 additive set is 𝒩 additive, thus we solve Problem 20 from paper [Weiss T., A note on the intersection ideal 𝒩 , Comment. Math. Univ. Carolin. 54 (2013), no. 3, 437-445] in the negative.

On the Behavior of Power Series with Completely Additive Coefficients

Oleg Petrushov (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Consider the power series ( z ) = n = 1 α ( n ) z , where α(n) is a completely additive function satisfying the condition α(p) = o(lnp) for prime numbers p. Denote by e(l/q) the root of unity e 2 π i l / q . We give effective omega-estimates for ( e ( l / p k ) r ) when r → 1-. From them we deduce that if such a series has non-singular points on the unit circle, then it is a zero function.

On strong measure zero subsets of κ 2

Aapo Halko, Saharon Shelah (2001)

Fundamenta Mathematicae

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We study the generalized Cantor space κ 2 and the generalized Baire space κ κ as analogues of the classical Cantor and Baire spaces. We equip κ κ with the topology where a basic neighborhood of a point η is the set ν: (∀j < i)(ν(j) = η(j)), where i < κ. We define the concept of a strong measure zero set of κ 2 . We prove for successor κ = κ < κ that the ideal of strong measure zero sets of κ 2 is κ -additive, where κ is the size of the smallest unbounded family in κ κ , and that the generalized Borel...

Sums of reciprocals of additive functions running over short intervals

J.-M. De Koninck, I. Kátai (2007)

Colloquium Mathematicae

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Letting f(n) = A log n + t(n), where t(n) is a small additive function and A a positive constant, we obtain estimates for the quantities x n x + H 1 / f ( Q ( n ) ) and x p x + H 1 / f ( Q ( p ) ) , where H = H(x) satisfies certain growth conditions, p runs over prime numbers and Q is a polynomial with integer coefficients, whose leading coefficient is positive, and with all its roots simple.

Ultrafilter extensions of asymptotic density

Jan Grebík (2019)

Commentationes Mathematicae Universitatis Carolinae

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We characterize for which ultrafilters on ω is the ultrafilter extension of the asymptotic density on natural numbers σ -additive on the quotient boolean algebra 𝒫 ( ω ) / d 𝒰 or satisfies similar additive condition on 𝒫 ( ω ) / fin . These notions were defined in [Blass A., Frankiewicz R., Plebanek G., Ryll-Nardzewski C., A Note on extensions of asymptotic density, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3313–3320] under the name A P (null) and A P (*). We also present a characterization of a P - and semiselective...

Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

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We establish an explicit connection between the perimeter measure of an open set E with C 1 boundary and the spherical Hausdorff measure S Q 1 restricted to E , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and Q denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of E is less than or equal to S Q 1 ( E ) up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

Relations between Shy Sets and Sets of ν p -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a non-zero non-atomic translation-invariant Borel measure ν p on the Banach space p ( 1 p ) is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " ν p -almost every element of p has a property P" implies that “almost every” element of p (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

On Ordinary and Standard Lebesgue Measures on

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on for α = (1,1,...).

Extensions of generic measure-preserving actions

Julien Melleray (2014)

Annales de l’institut Fourier

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We show that, whenever Γ is a countable abelian group and Δ is a finitely-generated subgroup of Γ , a generic measure-preserving action of Δ on a standard atomless probability space ( X , μ ) extends to a free measure-preserving action of Γ on ( X , μ ) . This extends a result of Ageev, corresponding to the case when Δ is infinite cyclic.

Interpolating sequences, Carleson measures and Wirtinger inequality

Eric Amar (2008)

Annales Polonici Mathematici

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Let S be a sequence of points in the unit ball of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure μ S : = a S ( 1 - | a | ² ) δ a is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of such that any δ -separated sequence S has its associated measure μ S bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of . As an easy consequence, we prove that if S is a dual...

Radon-Nikodym property

Surjit Singh Khurana (2017)

Commentationes Mathematicae Universitatis Carolinae

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For a Banach space E and a probability space ( X , 𝒜 , λ ) , a new proof is given that a measure μ : 𝒜 E , with μ λ , has RN derivative with respect to λ iff there is a compact or a weakly compact C E such that | μ | C : 𝒜 [ 0 , ] is a finite valued countably additive measure. Here we define | μ | C ( A ) = sup { k | μ ( A k ) , f k | } where { A k } is a finite disjoint collection of elements from 𝒜 , each contained in A , and { f k } E ' satisfies sup k | f k ( C ) | 1 . Then the result is extended to the case when E is a Frechet space.