an integrable function that does not tend to zero
and . Let , so is just a “shifted” version of . Note that
We now construct our function by defining . There are no convergence problems with this sum since for a given , at most one takes a non-zero value at . Also does not tend to 0 as as there are arbitrarily large values of for which takes the value , by (2).
All that is left is to show that is Lebesgue integrable. To do this rigorously, we apply the monotone convergence theorem (MCT) with . We must check the hypotheses of the MCT. Clearly as , and the sequence is monotone increasing, positive, and integrable. Furthermore, each is continuous and zero except on a compact interval, so is integrable. Finally, from (1) we see that for all . Therefore, the MCT applies and is integrable.
|Title||an integrable function that does not tend to zero|
|Date of creation||2013-03-22 16:56:09|
|Last modified on||2013-03-22 16:56:09|
|Last modified by||silverfish (6603)|