Real Analysis Exchange

The Asymptotic Behavior of Integrable Functions

Constantin Niculescu and Florin Popovici

Full-text: Open access


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

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

First available in Project Euclid: 29 April 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Lebesgue integral density convergence in density


Niculescu, Constantin; Popovici, Florin. The Asymptotic Behavior of Integrable Functions. Real Anal. Exchange 38 (2012), no. 1, 157--168.

Export citation


  • I. Barbălat, Systemes d'équations différentielle d'oscillations nonlinéaires, Rev. Roumaine Math. Pures Appl., 4 (1959), 267–270.
  • H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, New Jersey, 1981.
  • H. Halberstam and K. F. Roth, Sequences, 2nd Ed., Springer-Verlag, New York, 1983.
  • B. O. Koopman and J. von Neumann, Dynamical systems of continuous spectra, Proc. Natl. Acad. Sci. U.S.A., 18 (1932), 255–263.
  • E. Lesigne, On the Behavior at Infinity of an Integrable Function, Amer. Math. Monthly, 117 (2010), 175–181.
  • H. Logemann and E. P. Ryan, Asymptotic Behavior of Nonlinear Systems, Amer. Math. Monthly, 111 (2004), 864–889.
  • C. P. Niculescu and F. Popovici, A note on the behavior of integrable functions at infinity, J. Math. Anal. Appl., 381 (2011), 742–747.
  • T. Šalát and V. Toma, A classical Olivier's theorem and statistically convergence, Ann. Math. Blaise Pascal, 10 (2003), 305–313.