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