In [Univ. Beograd Publ. Elektrotehn. Fak. Ser. Math. 15 (2004), 85-86], the first author of this paper proved a new inequality for the Lebesgue measure and gave some applications. Here, we present a new application of this inequality. 

Aranđelović, Ivan

Petković, Dojčin S.

Kragujevac Journal of Mathematics

English

Machinery - Repository of the Faculty of Mechanical Engineering, University of Belgrade

53Kragujevac J. Math. 31 (2008) 53–57.AN INEQUALITY FOR THE LEBESGUE MEASUREAND ITS FURTHER APPLICATIONSIvan D. Arand-elović1 and Dojčin S. Petković21Faculty of Mechanical Engineering,Kraljice Marije 16, 11000 Beograd, Serbia(e-mail: iarandjelovic@mas.bg.ac.yu)2Faculty of Science and Mathematics,Knjaza Miloša 7, 28220 Kosovska Mitrovica, Serbia(Received May 26, 2008)Abstract. In [Univ. Beograd Publ. Elektrotehn. Fak. Ser. Math. 15 (2004), 85–86], thefirst author of this paper proved a new inequality for the Lebesgue measure and gave someapplications. Here, we present a new application of this inequality.1. INTRODUCTIONIf λ is the Lebesgue measure on the set of real numbers R and {An} sequences ofLebesgue measurable sets in R, then we have the following inequality:λ(limAn) ≤ limλ(An).But for the inequalitylimλ(An) ≤ λ(limAn)54we must suppose that λ(∪∞i=nAn) < ∞ for at least one value of n (see [7], p 40.).Example: for a family of intervals In = [n, n+1), n = 0, 1, . . ., we have: limλ(An) = 1and λ(limAn) = 0.In [3] first author present the following inequality, and as its applications shortand simple proofs of two famous Steinhaus’ results.Proposition 1. Let A be a measurable set of a positive measure and {xn} abounded sequence of real numbers. Thenλ(A) ≤ λ(lim(xn + A)).Further applications of this inequality was given in [4].2. RESULTSNow we prove the following additive form of the Aljančić -Arand-elović [1] Uniformconvergence theorem, which is one the fundamental results in the theory of O -regularly varying functions (see also [2], [5] or [6]).Proposition 2. (S. Aljančić and D. Arand-elović [1] ) Let f, g : R → R bea measurable functions such thatlims→∞(f(t + s)− f(s)) = g(t) < +∞,for all t ∈ R. Thenlims→∞ supt∈[a,b](f(t + s)− f(s)) < +∞,for any a, b ∈ R, (a < b).Proof. Letgr(t) = supx≥r(f(t + x)− f(x)).Hencelimr→∞ gr(t) = g(t) < +∞55for all t ∈ R. By Egoroff’s theorem follows that for any a, b ∈ R (a < b), there existsmeasurable set A ⊆ [a, b] such that λ(A) > 0 and convergence is uniform on A. Solims→∞ supt∈A(f(t + s)− f(s)) < +∞.Assume now that convergence is not uniform on [a, b]. Then there exists {xn} ⊆ [a, b]and {yn} ⊆ R such that lim yn = ∞ andlim(f(xn + yn)− f(yn)) = ∞.By Proposition 1, it follows thatλ(lim(A− xn)) ≥ λ(A) > 0,which implies that there exists t ∈ R and subsequence {xnj} ⊆ {xn} such that{t + xnj} ⊆ A. Then|f(xnj + ynj)− f(ynj)| ≤ |f(xnj + t + ynj − t)− f(ynj − t)|+ |f(ynj − t)− f(ynj)|.Nowlim|f(xnj + t + ynj − t)− f(ynj − t)| < ∞,because {t + xnj} ⊆ A and lim(ynj − t) = ∞. Fromlim(f(ynj − t)− f(ynj)) < ∞followslim(f(xn + yn)− f(yn)) < ∞,which is a contradiction. 23. ADDITIONAL COMMENTSCondition g is measurable can not be omitted because it apply Egoroff’s theoremto function g which introduce assumption that g is measurable.56Proposition 3. There exists measurable function f : R → R such thatlims→∞(f(t + s)− f(s)) = g(t) < +∞and g is non measurable.Proof. Let A ⊆ R be a measurable set such that:1◦) A = −A;2◦) A + 4 = A;3◦) A− A is the non measurable set;4◦) A ∩ (A + 2) = ∅.For existence of such set see [8].Let ϕB denote characteristic function of set B. The function f : R → R definedbyf(x) = ϕA(x)− ϕA+2(x) =−1, x ∈ (A + 2);0, x 6∈ A ∪ (A + 2);1, x ∈ A,is measurable. But function g defined byg(t) = limx→∞(f(x + t)− f(x)),is non measurable because such is the set{t | g(t) = 2} = A− (A + 2) = A− A− 2.2References[1] S. Aljančić and D. Arand-elović, O − Regularly varying functions, Publ. Inst.Math. (Beograd) 22 (36) (1977), 5-22.[2] D. Arand-elović, O − Regularly variation and uniform convergence, Publ. Inst.Math. (Beograd) 48 (62) (1990), 25-40.57[3] I. Arand-elović, An inequality for the Lebesgue measure, Univ. Beog. Publ. Elek.Fak. ser. Math. 15 (2004) 85-86.[4] I. Arand-elović and D. Petković, An inequality for the Lebesgue measure and itsapplications, Facta Universitatis 22 (2007) 11 - 14.[5] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, CambridgeUniv. Press 1987.[6] J. L. Geluk and L. de Haan, Regular variation, extensions and Tauberian theo-rems, CWI Tract 40, Amsterdam 1987.[7] P. Halmos, Measure theory, Van Nonstrand, Princeton 1950.[8] L. A. Rubel, A pathological Lebesgue-measurable function, J. London Math. Soc.38 (1963) 1-4.

An inequality for the Lebesgue measure and its further applications

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An inequality for the Lebesgue measure and its further applications

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