The function $f(x) = \sin(x)/x$ is Riemann Integrable from $0$ to $\infty$, but it is not Lebesgue Integrable on that same interval. (Note, it is not absolutely Riemann Integrable.)
Why is it we restrict our definition of Lebesgue Integrability to absolutely integrable? Wouldn't it be better to extend our definition to include ALL cases where Riemann Integrability holds, and use the current definition as a corollary for when the improper integral is absolutely integrable?
Best Answer
Technically speaking, the function $\displaystyle{f(x) = \frac{\sin x}{x}}$ is not Riemann integrable on $(0, \infty)$, but rather improperly Riemann integrable on $(0, \infty)$.
The construction of the Riemann integral only works for bounded intervals. We can extend this construction to unbounded intervals like $(0, \infty)$, but that requires an additional limiting process. It is the first construction (Riemann integrals for bounded integrals) that the Lebesgue integral generalizes.