Real Analysis Exchange

The Asymptotic Behavior of Integrable Functions

Constantin Niculescu and Florin Popovici

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Abstract

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, \(d^{(0)}(A)=\lim_{r\rightarrow\infty}\frac{m\left( A\cap\lbrack0,r)\right) }{r}.\)

Article information

Source
Real Anal. Exchange, Volume 38, Number 1 (2012), 157-168.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.rae/1367265645

Mathematical Reviews number (MathSciNet)
MR3083203

Zentralblatt MATH identifier
1275.26020

Subjects
Primary: 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX]
Secondary: 37A45: Relations with number theory and harmonic analysis [See also 11Kxx]

Keywords
Lebesgue integral density convergence in density

Citation

Niculescu, Constantin; Popovici, Florin. The Asymptotic Behavior of Integrable Functions. Real Anal. Exchange 38 (2012), no. 1, 157--168. https://projecteuclid.org/euclid.rae/1367265645


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