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 integrableMathworldPlanetmath on [0,) but f(x) does not tend to zero as x.

First of all, let gn be the function sin(2nx)χ[0,π2n], where χI denotes the characteristic functionMathworldPlanetmathPlanetmathPlanetmath of the interval I. In other words, χ takes the value 1 on I and 0 everywhere else.

Let μ denote Lebesgue measureMathworldPlanetmath. An easy computation shows

gn𝑑μ=21-n, (1)

and gn(π2n+1)=1. Let hn(x)=gn(x-nπ), so hn is just a “shifted” version of gn. Note that

hn(nπ+π2n+1)=1. (2)

We now construct our function f by defining f=r=0hr. There are no convergence problems with this sum since for a given x, at most one hr takes a non-zero value at x. Also f(x) does not tend to 0 as x 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 theoremMathworldPlanetmath (MCT) with fn=r=0nhr. We must check the hypotheses of the MCT. Clearly fnf as n, and the sequenceMathworldPlanetmath (fn) is monotone increasing, positive, and integrable. Furthermore, each fn is continuousMathworldPlanetmathPlanetmath and zero except on a compact interval, so is integrable. Finally, from (1) we see that fn𝑑μ4 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