## 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.

### MSC:

 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

### Keywords:

Lebesgue integral; density; convergence in density

### Citations:

Zbl 1227.40002; Zbl 1259.26009
Full Text:

### References:

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