On de Haan's Uniform Convergence Theorem

Mathematica Moravica, Vol
2008 unpublished
In [Univ. Beograd Publ. Elektrotehn. Fak. Ser. Math. 15 (2004), 85-86], we proved a new inequality for the Lebesgue measure and gave some applications. Here, we present as it new application new short and simple proof of de Haan's uniform convergence theorem. A measurable function g : (0, +∞) → (0, +∞) is translational O-regularly varying if (1) lim s→∞ g(s + t) g(s) < +∞ for each t ∈ R. For properties and applications of this class of mappings see Tasković [5]. Let λ be a Lebesgue measure on
more » ... besgue measure on the set of real numbers R. In [1] we present the following inequality, and as its applications short and simple proofs of two famous Steinhaus' results. Proposition 1 (I. Aranđelović [1]). Let A be a measurable set of a positive measure and {x n } a bounded sequence of real numbers. Then λ(A) ≤ λ(lim(x n + A)). Now, as new application of Proposition 1, we present the following new short and simple proof of de Haan's uniform convergence theorem [4]. For applications of this result see [2],[3] or [4]. Proposition 2 (L. de Haan [4]). Let f, g : R → R be a measurable functions such that g(s) > 0 for any s, g is translational O-regularly varying function and lim s→∞ f (t + s) − f (s) g(s) < +∞, for all t ∈ R. Then lim s→∞ sup t∈[a,b] f (t + s) − f (s) g(s) < +∞, for any a, b ∈ R such that a < b. 2000 Mathematics Subject Classification. Primary: 26A12; Secondary: 28A05.
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