A non-measurable set

2009 ◽  
pp. 193-195
Author(s):  
John Franks
Keyword(s):  
Measurable Set

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.

A note on rough set and non-measurable set

2000 ◽  
Vol 45 (16) ◽  
pp. 1456-1458
Author(s):  
Jian Yu ◽  
Qiansheng Cheng
Keyword(s):  
Rough Set ◽  
Measurable Set

The Continuity of the Visibility Function on a Starshaped Set

1972 ◽  
Vol 24 (5) ◽  
pp. 989-992 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Euclidean Space ◽  
Sufficient Conditions ◽  
Outer Measure ◽  
Visibility Function ◽  
Starshaped Set ◽  
Measurable Set

The visibility function assigns to each point x of a fixed measurable set E in a Euclidean space En the Lebesgue outer measure of S(x), the set {y : rx + (1 — r)y ∊ E for every r in [0, 1]}.The purpose of this paper is to determine sufficient conditions for the continuity of the function on the interor of a starshaped set.

scholarly journals Inequalities relating different definitions of discrepancy

1974 ◽  
Vol 17 (1) ◽  
pp. 81-87 ◽  
Author(s):  
C. J. Smyth
Keyword(s):  
Convex Sets ◽  
Measurable Set ◽  
Image Position

LetP1, P2…, pN be N points in the unit s-dimensional closed square Q = [0, 1]s. For any measurable set S ⊆ Q, we define δ(S), the discrepancy of S, by , where V(S) is the s-dimensional volume of S, and n(S), is the number of indices i for which pi∈S. Let , where the supermum is taken over all s-balls B ∈ Q, and , the supermum in this case being taken over all convex sets C ∈ Q. Clearly Dc ≧ Dk. In this paper we establish Theorem , where φ1is a constant depending only on s.

scholarly journals Torsion Discriminance for Stability of Linear Time-Invariant Systems

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 386
Author(s):  
Yuxin Wang ◽  
Huafei Sun ◽  
Yueqi Cao ◽  
Shiqiang Zhang
Keyword(s):  
Lebesgue Measure ◽  
Linear Time ◽  
Zero Solution ◽  
Positive Lebesgue Measure ◽  
Asymptotically Stable ◽  
Linear Time Invariant ◽  
Measurable Set ◽  
Linear Time Invariant Systems ◽  
The Stability ◽  
Time Invariant

This paper extends the former approaches to describe the stability of n-dimensional linear time-invariant systems via the torsion τ ( t ) of the state trajectory. For a system r ˙ ( t ) = A r ( t ) where A is invertible, we show that (1) if there exists a measurable set E 1 with positive Lebesgue measure, such that r ( 0 ) ∈ E 1 implies that lim t → + ∞ τ ( t ) ≠ 0 or lim t → + ∞ τ ( t ) does not exist, then the zero solution of the system is stable; (2) if there exists a measurable set E 2 with positive Lebesgue measure, such that r ( 0 ) ∈ E 2 implies that lim t → + ∞ τ ( t ) = + ∞ , then the zero solution of the system is asymptotically stable. Furthermore, we establish a relationship between the ith curvature ( i = 1 , 2 , ⋯ ) of the trajectory and the stability of the zero solution when A is similar to a real diagonal matrix.

Group Actions and Singular Martingales II, The Recognition Problem

2004 ◽  
Vol 56 (2) ◽  
pp. 431-448
Author(s):  
Joseph Rosenblatt ◽  
Michael Taylor
Keyword(s):  
Inverse Problem ◽  
Compact Group ◽  
Harmonic Analysis ◽  
Group Actions ◽  
Gegenbauer Polynomials ◽  
Recognition Problem ◽  
Measurable Set

AbstractWe continue our investigation in [RST] of a martingale formed by picking a measurable set A in a compact group G, taking random rotates of A, and considering measures of the resulting intersections, suitably normalized. Here we concentrate on the inverse problem of recognizing A from a small amount of data from this martingale. This leads to problems in harmonic analysis on G, including an analysis of integrals of products of Gegenbauer polynomials.

scholarly journals Remarks to the Paper “On Montel’s Theorem’ By Kawakami

1956 ◽  
Vol 10 ◽  
pp. 165-169
Author(s):  
Makoto Ohtsuka
Keyword(s):  
Linear Measure ◽  
Measurable Set ◽  
Montel’S Theorem

We take a measurable set E on the positive η-axis and denote by μ(r) the linear measure of the part of E in the interval 0 < η < r. The lower density of E at η = 0 is defined byTheorem by Kawakami [1] asserts that if λ is positive, if a function f(ζ) = f(ξ + iη) is bounded analytic in ξ > 0 and continuous at E, and if f(ζ) → A as ζ → 0 along E, then f(ζ) → A as ζ → 0 in ∣η∣ ≦ kξ for any k > 0.

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

ON CONVERGENCE OF THE FOURIER DOUBLE SERIES WITH RESPECT TO THE VILENKIN SYSTEMS

2018 ◽  
Vol 52 (1 (245)) ◽  
pp. 12-18
Author(s):  
L.S. Simonyan
Keyword(s):  
Double Series ◽  
Natural Numbers ◽  
Vilenkin System ◽  
Measurable Set ◽  
Almost Everywhere

Let $ \{ W_k (x) \} _{k = 0}^{\infty} $ be either unbounded or bounded Vilenkin system. Then, for each $ 0 < \varepsilon < 1 $, there exist a measurable set $ E \subset [0,1)^2 $ of measure $ |E| > 1 \mathclose{-} \varepsilon $, and a subset of natural numbers $ \Gamma $ of density 1 such that for any function $ f(x,y) \in L^1 (E) $ there exists a function $ g(x,y) \in L^1 [0,1)^2 $, satisfying the following conditions: $ g(x, y) = f(x,y) $ on $ E \ ; $ the nonzero members of the sequence $ \{ |c_{k, s}(g)| \} $ are monotonically decreasing in all rays, where $ c_{k, s} (g) = \int\limits_{0}^{1} \int\limits_{0}^{1} g(x, y) \overline{W_k}(x) \overline{W_s}(y) dx dy \ ; $ $ \lim\limits_{R \in \Gamma,\ R \to \infty} S_R((x,y),g) = g(x,y) $ almost everywhere on $ [0,1)^2 $, where $ S_R((x,y),g) = \sum\limits_{k^2+s^2 \leq R^2} c_{k, s}(g) W_k(x) W_s(y) $.

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