Mostrando recursos 1 - 20 de 41

  1. A Note on Monotonicity Theorems for Approximately Continuous Functions

    Fejzić, Hajrudin
    A simple proof of a well-known monotonicity theorem for the approximate derivative is presented. The same technique provides a proof of a generalized version of Ornesteinʼs Theorem.

  2. Measure, Category and Convergent Series

    Labuda, Iwo
    The analogy between measure and Baire category is displayed first by a theorem of Steinhaus and its “dual,” a theorem of Piccard. These two theorems are then applied to provide a double criterion for the unconditional convergence of a series in terms of the “measure size” and the “category size“ of the set of its convergent subseries. As a further application, after a substantial preparatory section concerning essential separability of measurable and \(BP\)-measurable functions, the results about exhaustivity of \(BP_r\)-measurable and universally measurable additive maps on the Cantor group are established. In the last sections of the paper, two classical...

  3. On Some Gamidov Integral Inequalities on Time Scales and Applications

    Meftah, Badreddine
    In the present paper, we extend Gamidov’s integral inequalities to time scales. The obtained results can be used as tools in the study of certain properties of dynamical equations on time scales.

  4. A Positive Function with Vanishing Lebesgue Integral in Zermelo-Fraenkel Set Theory

    Kanovei, Vladimir; Katz, Mikhail
    Can a positive function on \(\mathbb{R}\) have zero Lebesgue integral? It depends on how much choice one has.

  5. On the Carathéodory Approach to the Construction of a Measure

    Werner, Ivan
    The Carathéodory theorem on the construction of a measure is generalized by replacing the outer measure with an approximation of it and generalizing the Carathéodory measurability. The new theorem is applied to obtain dynamically defined measures from constructions of outer measure approximations resulting from sequences of measurement pairs consisting of refining \(\sigma\)-algebras and measures on them which need not be consistent. A particular case when the measurement pairs are given by the action of an invertible map on an initial \(\sigma\)-algebra and a measure on it is also considered.

  6. Multiplication Operators on the Spaces of Functions of Bounded \(p\)-Variation in Wiener’s Sense

    Astudillo-Villaba, Franklin R.; Castillo, René E.; Ramos-Fernández, Julio C.
    In this article, we make a comprehensive study of the properties of multiplication operators acting on the spaces of functions of bounded \(p\)-variation in Wiener’s sense, \(WBV_p[0,1]\). We characterize all functions \(u\in WBV_p[0,1]\) that define invertible, compact and Fredholm multiplication operators \(M_u\) on \(WBV_p[0,1]\). Also we characterize when \(M_u\) has finite range and has closed range on \(WBV_p[0,1]\).

  7. On Exceptional Sets of the Hilbert Transform

    Karagulyan, Grigori A.
    We prove several theorems concerning the exceptional sets of the Hilbert transform on the real line. In particular, it is proved that any null set is an exceptional set for the Hilbert transform of an indicator function. The paper also provides a real variable approach to the Kahane-Katsnelson theorem on divergence of Fourier series.

  8. The McShane Integral in the Limit

    Sayyad, Redouane
    We introduce the notion of the McShane integral in the limit for functions defined on a \(\sigma\)-finite outer regular quasi Radon measure space \((S,\Sigma,\mathcal{T},\mu)\) into a Banach space \(X\) and we study its relation with the generalized McShane integral introduced by D. H. Fremlin. It is shown that if a function from \(S\) into \(X\) is McShane integrable in the limit on \(S\) and scalarly locally \(\tau\)-upper McShane bounded for some \(\tau >0\), then it is McShane integrable on \(S\). On the other hand, we prove that if an \(X\)-valued function is McShane integrable in the limit on \(S\), then...

  9. A Generalized Egorov’s Statement for Ideals

    Korch, Mikhail
    We consider the generalized Egorov’s statement (Egorov’s Theorem without the assumption on measurability of the functions, see Weiss (2004)) in the case of an ideal convergence and a number of different types of ideal convergence notions. We prove that in those cases the generalized Egorov’s statement is independent from ZFC.

  10. Some Results in Support of the Kakeya Conjecture

    Fraser, Jonathan M.; Olson, Eric J.; Robinson, James C.
    A Besicovitch set is a subset of \(\mathbb{R}^d\) that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this conjecture in a variety of contexts. Our proofs are simple and aim to give an intuitive feel for the problem. For example, we give a very simple proof that the packing and lower box-counting dimension of any Besicovitch set is at least \((d+1)/2\) (better estimates are available in the literature). We also study the “generic validity” of the Kakeya conjecture...

  11. Approaches to Analysis with Infinitesimals Following Robinson, Nelson, and Others

    Fletcher, Peter; Hrbacek, Karel; Kanovei, Vladimir; Katz, Mikhail G.; Lobry, Claude; Sanders, Sam
    This is a survey of several approaches to the framework for working with infinitesimals and infinite numbers, originally developed by Abraham Robinson in the 1960s, and their constructive engagement with the Cantor-Dedekind postulate and the Intended Interpretation hypothesis. We highlight some applications including (1) Loeb’s approach to the Lebesgue measure, (2) a radically elementary approach to the vibrating string, (3) true infinitesimal differential geometry. We explore the relation of Robinson’s and related frameworks to the multiverse view as developed by Hamkins.

  12. Which Integrable Functions Fail to be Absolutely Integrable?

    Mendoza, José
    An answer to the question of the title is given.

  13. A note on the Luzin-Menchoff theorem

    Fejzić, Hajrudin
    A proof of the Luzim-Menchoff theorem.

  14. On the Minkowski Sum of Two Curves

    Bruckner, Andrew M.; Ciesielski, Krzysztof Chris
    We show that there exists a derivative \(f\colon [0,1]\to[0,1]\) such that the graph of \(f\circ f\) is dense in \([0,1]^2\), so not a \(G_\delta\)-set. In particular, \(f\circ f\) is everywhere discontinuous, so not of Baire class 1, and hence it is not a derivative. %neither of Baire class 1 nor a derivative.

  15. A Note on the Uniqueness Property for Borel G-measures

    Kharazishvili, Alexander
    In terms of a group \(G\) of isometries of Euclidean space, it is given a necessary and sufficient condition for the uniqueness of a \(G\)-measure on the Borel \(\sigma\)-algebra of this space.

  16. On the Minkowski Sum of Two Curves

    Chang, Alan
    We answer a question posed by Miklós Laczkovich on the Minkowski sum of two curves.

  17. Random Cutouts of the Unit Cube with I.U.D Centers

    Zhu, Z. Y.; Dong, E. M.
    Consider the random open balls \(B_n(\omega):=B(\omega_n,r_n)\) with their centers \(\omega_n\) being i.u.d. on the \(d\)-dimensional unit cube \([0,1]^d\) and with their radii \(r_n\sim cn^{-\frac{1}{d}}\) for some constant \(0

  18. Magic Sets

    Halbeisen, Lorenz; Lischka, Marc; Schumacher, Salome
    In this paper we study magic sets for certain families \(\mathcal{H}\subseteq {^\mathbb{R}\mathbb{R}}\) which are subsets \(M\subseteq\mathbb{R}\) such that for all functions \(f,g\in\mathcal{H}\) we have that \(g[M]\subseteq f[M]\Rightarrow f=g\). Specifically we are interested in magic sets for the family \(\mathcal{G}\) of all continuous functions that are not constant on any open subset of \(\mathbb{R}\). We will show that these magic sets are stable in the following sense: Adding and removing a countable set does not destroy the property of being a magic set. Moreover, if the union of less than \(\mathfrak{c}\) meager sets is still meager, we can also add and...

  19. Divided Differences, Square Functions, and a Law of the Iterated Logarithm

    Nicolau, Artur
    The main purpose of the paper is to show that differentiability properties of a measurable function defined in Euclidean space can be described using square functions which involve its second symmetric divided differences. Classical results of Marcinkiewicz, Stein and Zygmund describe, up to sets of Lebesgue measure zero, the set of points where a function \(f\) is differentiable in terms of a certain square function \(g(f)\). It is natural to ask for the behavior of the divided differences at the complement of this set, that is, on the set of points where \(f\) is not differentiable. In the nineties, Anderson...

  20. Ergodic Properties of Rational Functions that Preserve Lebesgue Measure on ℝ

    Bayless, Rachel L.
    We prove that all negative generalized Boole transformations are conservative, exact, pointwise dual ergodic, and quasi-finite with respect to Lebesgue measure on the real line. We then provide a formula for computing the Krengel, Parry, and Poisson entropy of all conservative rational functions that preserve Lebesgue measure on the real line.

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