TITLE: Is every Riemann-integrable function lebesgue integrable?
QUESTION [7 upvotes]: I thought every Riemann integrable function is lebesgue integrable. But here, a user comments that the statement is not correct.  I'm confused.  Can someone clarify?

REPLY [3 votes]: No. This does not have to always hold on unbounded intervals. 
$$\int_{0}^{\infty} \frac{\sin{x}}{x} \, dx$$ exists in the sense of Riemann, but does not exist in the sense of Lebesgue as 
$$\int_{(0,\infty)} \frac{\sin{x}}{x} \, d\mu. $$
This has to do with the fact that there is no "conditional convergence" in Lebesgue Integration.