Real Analysis Exchange

The Asymptotic Behavior of Integrable Functions

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

https://projecteuclid.org/euclid.rae/1367265645

Mathematical Reviews number (MathSciNet)
MR3083203

Zentralblatt MATH identifier
1275.26020

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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