What is the relationship between the improper Riemann-integral and Lebesgue integrability?
There's a very sloppy section of it in my book with about 4-5 known mistakes found so far, so I fear I have a completely wrong idea of what's what now.
My question is, what must be true for some $u$ and its improper Riemann-integral for it to be Lebesgue-integrable, and what must be true for the Lebesgue-integrability of $u$ for it to have finite or infinite improper Riemann-integrable?
I know for example that $\sin(x) / x$, $x \in (0,\infty)$ is improperly Riemann-integrable but not Lebesgue integrable. What's gone wrong here?