A non-measurable set

Author(s):  
John Franks
Keyword(s):  
2020 ◽  
Vol 70 (2) ◽  
pp. 305-318
Author(s):  
Anna Kamińska ◽  
Katarzyna Nowakowska ◽  
Małgorzata Turowska

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.


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

1972 ◽  
Vol 24 (5) ◽  
pp. 989-992 ◽  
Author(s):  
Gerald Beer

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.


1974 ◽  
Vol 17 (1) ◽  
pp. 81-87 ◽  
Author(s):  
C. J. Smyth

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.


Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 386
Author(s):  
Yuxin Wang ◽  
Huafei Sun ◽  
Yueqi Cao ◽  
Shiqiang Zhang

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.


2004 ◽  
Vol 56 (2) ◽  
pp. 431-448
Author(s):  
Joseph Rosenblatt ◽  
Michael Taylor

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.


1956 ◽  
Vol 10 ◽  
pp. 165-169
Author(s):  
Makoto Ohtsuka

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.


1991 ◽  
Vol 33 (2) ◽  
pp. 129-134
Author(s):  
Szilárd GY. Révész ◽  
Imre Z. Ruzsa

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


2018 ◽  
Vol 52 (1 (245)) ◽  
pp. 12-18
Author(s):  
L.S. Simonyan

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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