# an integrable function that does not tend to zero

In this entry, we give an example of a function $f$ such that $f$ is Lebesgue integrable^{} on $[0,\mathrm{\infty})$ but $f(x)$ does not tend to zero as $x\to \mathrm{\infty}$.

First of all, let ${g}_{n}$ be the function $\mathrm{sin}({2}^{n}x){\chi}_{[0,\frac{\pi}{{2}^{n}}]}$, where ${\chi}_{I}$ denotes the characteristic function^{} of the interval $I$. In other words, $\chi $ takes the value $1$ on $I$ and 0 everywhere else.

Let $\mu $ denote Lebesgue measure^{}. An easy computation shows

$${\int}_{\mathbb{R}}{g}_{n}\mathit{d}\mu ={2}^{1-n},$$ | (1) |

and ${g}_{n}\left({\displaystyle \frac{\pi}{{2}^{n+1}}}\right)=1$. Let ${h}_{n}(x)={g}_{n}(x-n\pi )$, so ${h}_{n}$ is just a “shifted” version of ${g}_{n}$. Note that

$${h}_{n}\left(n\pi +\frac{\pi}{{2}^{n+1}}\right)=1.$$ | (2) |

We now construct our function $f$ by defining $f={\displaystyle \sum _{r=0}^{\mathrm{\infty}}}{h}_{r}$. There are no convergence problems with this sum since for a given $x\in \mathbb{R}$, at most one ${h}_{r}$ takes a non-zero value at $x$. Also $f(x)$ does not tend to 0 as $x\to \mathrm{\infty}$ as there are arbitrarily large values of $x$ for which $f$ takes the value $1$, by (2).

All that is left is to show that $f$ is Lebesgue integrable. To do this rigorously, we apply the monotone convergence theorem^{} (MCT) with ${f}_{n}={\displaystyle \sum _{r=0}^{n}}{h}_{r}$. We must check the hypotheses of the MCT. Clearly ${f}_{n}\to f$ as $n\to \mathrm{\infty}$, and the sequence^{} $({f}_{n})$ is monotone increasing, positive, and integrable. Furthermore, each ${f}_{n}$ is continuous^{} and zero except on a compact interval, so is integrable. Finally, from (1) we see that ${\int}_{\mathbb{R}}}{f}_{n}\mathit{d}\mu \le 4$ for all $n$. Therefore, the MCT applies and $f$ is integrable.

Title | an integrable function that does not tend to zero |
---|---|

Canonical name | AnIntegrableFunctionThatDoesNotTendToZero |

Date of creation | 2013-03-22 16:56:09 |

Last modified on | 2013-03-22 16:56:09 |

Owner | silverfish (6603) |

Last modified by | silverfish (6603) |

Numerical id | 9 |

Author | silverfish (6603) |

Entry type | Example |

Classification | msc 28-01 |