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I'm trying to prove the following:

Show that the integral of an absolutely continuous function, which asymptotically converges to zero (so the function value is zero when its argument is infinity), exists. In other words, I need to show that the integral tends to some constant $c<\infty$.

Any help is greatly appreciated!

Edit It's exactly the opposite of this problem.

Community
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rbrns
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  • What's the defintion of an absolutely continuous function on an unbounded interval? – zhw. Oct 24 '16 at 20:10
  • I assumed that the definition of absolute continuity on a closed interval can be extended to a (half) open inteval, as discussed in this post. Actually, I'm not a 100% sure if this assumption is justifiable. – rbrns Oct 25 '16 at 07:48
  • Maybe assume the (AC) function is of the form $\int _a ^x f(t)\mu(\mathrm{d}t)$? That is, it is a Lebesgue's integral. – Ranc Oct 25 '16 at 07:58

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This is clearly not true, as $x \mapsto \frac{1}{x + 1}$ is absolutely continuous over $[0, +\infty[$, tends to zero but has no definite integral.

Ulysse Mizrahi
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  • An excellent example, thanks. But are there conditions that the function should satisfy in order for the limit of the integral to exist? – rbrns Oct 25 '16 at 09:31
  • well, given that any continuous function with limit 0 at infinity is absolutely continuous, the conditions are the same as for any continuous function with limit 0. – Ulysse Mizrahi Oct 25 '16 at 09:38
  • Any idea where I can find those specific conditions? Thanks for your answer. – rbrns Oct 25 '16 at 09:56
  • There are no simple conditions unfortunately other than just "be integrable". Also consider accepting the answer if it fits your question. – Ulysse Mizrahi Oct 25 '16 at 11:52
  • I understand. Thank you for your effort. – rbrns Oct 25 '16 at 12:20