The asymptotic behavior of integrable functions. (English) Zbl 1275.26020

If \(f:[0,\infty)\to\mathbb R\) is a function such that the Lebesgue integral \(\int_0^\infty f\) exists and is finite, then it is not necessarily true that \(\lim_{x\to\infty}f(x)=0\). The statement becomes true if we either strengthen the assumptions on \(f\) (for example, the proposition holds for nonnegative nonincreasing functions, as well as for uniformly continuous functions), or if we replace the usual limit by a weaker concept such as the limit in density.
Given a density function \(d\) defined on the Borel subsets of \([0,\infty)\), we say that the limit of \(f\) at infinity in density is zero if \(d(\{t;\,|f(t)|\geq\varepsilon\})=0\) for every \(\varepsilon>0\). In this case, we write \[ (d)-\lim_{x\to\infty}f(x)=0. \]
In their previous paper [J. Math. Anal. Appl. 381, No. 2, 742–747 (2011; Zbl 1227.40002)], the authors have considered the densities \(d_0\), \(d_1\) given by \[ d_0(A)=\lim_{r\to\infty}{1\over r}\int_{A\cap[0,r)}\,{\mathrm d}t, \]
\[ d_1(A)=\lim_{r\to\infty}{1\over \ln r}\int_{A\cap[1,r)}\,{{\mathrm d}t\over t}, \] and proved that \((d_0)-\lim_{x\to\infty}x f(x)=0\) and \((d_1)-\lim_{x\to\infty}(x\ln x)f(x)=0\) for every Lebesgue integrable function \(f:[0,\infty)\to\mathbb R\).
The main result of the present paper is the following more general statement: Every Lebesgue integrable function \(f:[0,\infty)\to\mathbb R\) satisfies \[ (d_n)-\lim_{x\to\infty}\left(\prod_{k=0}^n\ln^{(k)}x\right)f(x)=0,\quad n\in\mathbb N,\eqno(1) \] where \(d_n\) is the density given by \[ d_n(A)=\lim_{r\to\infty}{1\over \ln^{(n)} r}\int_{A\cap[\exp^{(n-1)}1,r)}\,{{\mathrm d}t\over \prod_{k=0}^{n-1}\ln^{(k)}t},\eqno(2) \] and \(g^{(k)}\) stands for the \(k\)-th iterate of a function \(g\).
The authors also show that for any density given by (2) and every measurable function \(f:[0,\infty)\to\mathbb R\), the convergence in density to zero is equivalent to the existence of a set \(S\subset[0,\infty)\) such that \(d_n(A)=0\) and \[ \lim_{x\to\infty,\,x\notin S}f(x)=0. \]
Reviewer’s remark: The main result (1) was already proved in the paper [Expo. Math. 30, No. 3, 277–282 (2012; Zbl 1259.26009)] by the same authors.


26A42 Integrals of Riemann, Stieltjes and Lebesgue type
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
40A10 Convergence and divergence of integrals
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