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2012/2013 The Asymptotic Behavior of Integrable Functions
Constantin Niculescu, Florin Popovici
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Real Anal. Exchange 38(1): 157-168 (2012/2013).


Given a density \(d\) defined on the Borel subsets of \([0,\infty),\) the limit in density of a function \(f:[0,\infty)\rightarrow\mathbb{R}\) is zero (abbreviated, \((d)\)-\(\lim_{x\rightarrow\infty}f(x)=0)\) if there exists a set \(S\) of zero density such that \(f(x)\rightarrow0\) as \(x\ \)runs to \(\infty\) outside \(S\). It is proved that the behavior at infinity of every Lebesgue integrable function \(f\in L^{1}(0,\infty)\) satisfies the relations \[ (d^{(n)})-\lim_{x\rightarrow\infty}\left( \prod\nolimits_{k=0}^{n}\ln ^{(k)}x\right) f(x)=0, \] where \((d^{(n)})_{n}\) is a scale of densities including the usual one, \(d^{(0)}(A)=\lim_{r\rightarrow\infty}\frac{m\left( A\cap\lbrack0,r)\right) }{r}.\)


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Constantin Niculescu. Florin Popovici. "The Asymptotic Behavior of Integrable Functions." Real Anal. Exchange 38 (1) 157 - 168, 2012/2013.


Published: 2012/2013
First available in Project Euclid: 29 April 2013

zbMATH: 1275.26020
MathSciNet: MR3083203

Primary: 26A42
Secondary: 37A45

Keywords: convergence in density , Density , Lebesgue integral

Rights: Copyright © 2012 Michigan State University Press

Vol.38 • No. 1 • 2012/2013
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