On sets of points of approximate continuity and ϱ-upper continuity

2020 ◽  
Vol 70 (2) ◽  
pp. 305-318
Author(s):  
Anna Kamińska ◽  
Katarzyna Nowakowska ◽  
Małgorzata Turowska
Keyword(s):  
Real Function ◽  
Approximate Continuity ◽  
Measurable Set ◽  
Upper Continuous ◽  
Lebesgue Measurable Set ◽  
Upper Continuity

Abstract In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

scholarly journals Measurability of functions of two variables

1959 ◽  
Vol 1 (1) ◽  
pp. 21-26 ◽  
Author(s):  
J. H. Michael ◽  
B. C. Rennie
Keyword(s):  
Real Function ◽  
The Other ◽  
Main Result Theorem ◽  
Functions Of Two Variables ◽  
Measurable Set ◽  
Result Theorem ◽  
Lebesgue Measurable Set

SummaryThis paper investigates the existence and equality of the double and repeated integrals of a real function on a plane set. The main result (Theorem 2) is that if a function on a plane Lebesgue measurable set is continuous in one variable and measurable in the other then it is measurable in the plane.

scholarly journals On approximating Lebesgue integrals by Riemann sums

1991 ◽  
Vol 33 (2) ◽  
pp. 129-134
Author(s):  
Szilárd GY. Révész ◽  
Imre Z. Ruzsa
Keyword(s):  
Characteristic Function ◽  
Lebesgue Measure ◽  
Real Function ◽  
Riemann Sums ◽  
Function Classes ◽  
Open Set ◽  
Measurable Set ◽  
Image Position ◽  
Good Sequences

If f is a real function, periodic with period 1, we defineIn the whole paper we write ∫ for , mE for the Lebesgue measure of E ∩ [0,1], where E ⊂ ℝ is any measurable set of period 1, and we also use XE for the characteristic function of the set E. Consistent with this, the meaning of ℒp is ℒp [0, 1]. For all real xwe haveif f is Riemann-integrable on [0, 1]. However,∫ f exists for all f ∈ ℒ1 and one would wish to extend the validity of (2). As easy examples show, (cf. [3], [7]), (2) does not hold for f ∈ ℒp in general if p < 2. Moreover, Rudin [4] showed that (2) may fail for all x even for the characteristic function of an open set, and so, to get a reasonable extension, it is natural to weaken (2) towhere S ⊂ ℕ is some “good” increasing subsequence of ℕ. Naturally, for different function classes ℱ ⊂ ℒ1 we get different meanings of being good. That is, we introduce the class of ℱ-good sequences as

scholarly journals Pointwise density topology

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Magdalena Górajska
Keyword(s):  
Real Line ◽  
Pointwise Convergence ◽  
Baire Property ◽  
Density Topology ◽  
Borel Set ◽  
Convergence In Measure ◽  
The Real ◽  
Measurable Set ◽  
New Type ◽  
Lebesgue Measurable Set

AbstractThe paper presents a new type of density topology on the real line generated by the pointwise convergence, similarly to the classical density topology which is generated by the convergence in measure. Among other things, this paper demonstrates that the set of pointwise density points of a Lebesgue measurable set does not need to be measurable and the set of pointwise density points of a set having the Baire property does not need to have the Baire property. However, the set of pointwise density points of any Borel set is Lebesgue measurable.

Loci of complex polynomials, part II: polar derivatives

2015 ◽  
Vol 159 (2) ◽  
pp. 253-273 ◽  
Author(s):  
BLAGOVEST SENDOV ◽  
HRISTO SENDOV
Keyword(s):  
Higher Order ◽  
Complex Polynomial ◽  
Complex Polynomials ◽  
Point Sets ◽  
Closed Set ◽  
Measurable Set ◽  
Closed Point ◽  
Polar Derivatives ◽  
Lebesgue Measurable Set ◽  
Derivatives Of

AbstractFor every complex polynomial p(z), closed point sets are defined, called loci of p(z). A closed set Ω ⊆ ${\mathbb C}$* is a locus of p(z) if it contains a zero of any of its apolar polynomials and is the smallest such set with respect to inclusion. Using the notion locus, some classical theorems in the geometry of polynomials can be refined. We show that each locus is a Lebesgue measurable set and investigate its intriguing connections with the higher-order polar derivatives of p.

scholarly journals On the p-norm of the truncated Hilbert transform

1988 ◽  
Vol 38 (3) ◽  
pp. 413-420 ◽  
Author(s):  
W. McLean ◽  
D. Elliott
Keyword(s):  
Hilbert Transform ◽  
Positive Measure ◽  
Integral Operators ◽  
Finite Hilbert Transform ◽  
Measurable Set ◽  
Symmetry Properties ◽  
Lebesgue Measurable Set ◽  
The One ◽  
The Hilbert Transform

The p-norm of the Hilbert transform is the same as the p-norm of its truncation to any Lebesgue measurable set with strictly positive measure. This fact follows from two symmetry properties, the joint presence of which is essentially unique to the Hilbert transform. Our result applies, in particular, to the finite Hilbert transform taken over (−1, 1), and to the one-sided Hilbert transform taken over (0, ∞). A related weaker property holds for integral operators with Hardy kernels.

On the lack of equi-measurability for certain sets of Lebesgue-measurable functions

2017 ◽  
Vol 67 (6) ◽  
Author(s):  
Marianna Tavernise ◽  
Alessandro Trombetta ◽  
Giulio Trombetta
Keyword(s):  
Measurable Functions ◽  
Measurable Set ◽  
Lebesgue Measurable Set

AbstractLet Ω be a Lebesgue-measurable set in ℝ

Some sets with measurable inverses

1969 ◽  
Vol 65 (2) ◽  
pp. 437-438
Author(s):  
Roy O. Davies
Keyword(s):  
Real Function ◽  
Continuum Hypothesis ◽  
Axiom Of Choice ◽  
Borel Set ◽  
Analytic Set ◽  
Linear Set ◽  
Measurable Set ◽  
Intermediate Results ◽  
The Axiom Of Choice ◽  
The Continuum

Goldman (4) conjectured that if Z is a linear set having the property that for every (Lebesgue) measurable real function f the set f−1[Z] is a measurable set, then Z must be a Borel set. I pointed out (2) that any analytic non-Borel set provides a counterexample, and Eggleston(3) showed that a set can have the property but be neither analytic nor even an analytic complement, for example, any Luzin set. As Eggleston mentions, in the construction of Luzin sets the continuum hypothesis is assumed (compare Sierpiński(6), Chapter II), and the question arises whether it can be dispensed with in his theorem. We shall show that a non-analytic set having Goldman's property can be constructed with the help of the axiom of choice alone, without the continuum hypothesis; the problem for analytic complements remains open. We shall also generalize one of Eggleston's intermediate results.

Quantitative continuation from a measurable set of solutions of elliptic equations

2000 ◽  
Vol 130 (4) ◽  
pp. 909-923 ◽  
Author(s):  
S. Vessella
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Positive Measure ◽  
Symmetric Matrix ◽  
Measurable Set ◽  
Connected Set ◽  
Lebesgue Measurable Set

Consider an open bounded connected set Ω in Rn and a Lebesgue measurable set E ⊂⊂ Ω of positive measure. Let u be a solution of the strictly elliptic equation Di (aij Dj u) = 0 in Ω, where aij ∈ C0, 1 (Ω̄) and {aij} is a symmetric matrix. Assume that |u| ≤ ε in E. We quantify the propagation of smallness of u in Ω.

Perturbation of functions by the paths of a Lévy process

1989 ◽  
Vol 105 (2) ◽  
pp. 377-380 ◽  
Author(s):  
Steven N. Evans
Keyword(s):  
Continuous Function ◽  
Lebesgue Measure ◽  
Stable Process ◽  
Levy Process ◽  
Probabilistic Argument ◽  
Scaling Properties ◽  
Measurable Set ◽  
Lebesgue Measurable Set ◽  
Image Position ◽  
Intermediate Value

In a recent paper Mountford [4] showed, using an ingenious probabilistic argument, that if X is a real-valued stable process with index α < 1 and f: [0, ∞) → ℝ is a non-constant continuous function, thenwhere we use the notation |A| for the Lebesgue measure of a Lebesgue measurable set A ⊂ ℝ. The argument in [4] appears to make strong use of the strict scaling properties of X and the ‘intermediate value’ property of f.

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