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I happened upon the following integration exercise. I don't know if it would count as standard in measure theory, but it seems interesting, and I am wondering how best to approach it.

Let $r \in \mathbb{R}$. For which values of $r$ is the following function from $(0,1] \rightarrow \mathbb{R}$ (1) Riemann integrable in the improper sense or (2) Lebesgue integrable:

$$y^r \sin(1/y).$$

For (1), I was thinking that an effective argument will require integration by parts and L'Hopital's Rule in tandem, but I am a little rusty in using these tools (it's embarassing!); perhaps the Riemann-Lebesgue lemma will factor in too. Unfortunately, I don't have such a "knee-jerk response" in mind for (2), since Lebesgue integration is something new that I haven't fully grappled with yet. Any assistance a visitor to the site cares to give would be appreciated! Thanks.

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    I don't see why the Riemann-Lebesgue lemma should factor in. What do you mean specifically? (Especially since the RL lemma is a statement where the hypothesis requires the function to be integrable.)2012-03-02

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