Why do we restrict the definition of Lebesgue Integrability? 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?

I'll add an additional answer based on a response from my measure theory professor, as it may be of use.
Essentially, his response was that the purpose of Lebesgue Integration is to make the set of integrable functions complete.  (Recall that we can form a sequence of Riemann Integrable functions that converge to a function that is not Riemann Integrable.)  As was noted by @Carl Mummert in his answer, the things of interest in Lebesgue Theory are convergence theorems (which tie into this notion of completeness), not a theory of integrals for specific classes of functions.
As such, it isn't that the definition of Lebesgue Integrability isn't as broad as it might be, but that it is broad enough to ensure completeness.
