TITLE: Why do we restrict the definition of Lebesgue Integrability?
QUESTION [45 upvotes]: 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?

REPLY [4 votes]: If one thinks about an integral as giving you the area under a curve, this area should be expected to behave in a natural fashion, ie, if you chop it up into pieces and rearrange them via translations, the area should be preserved.
This property will hold for any $L^1$ function, but it need not be true for a function which has only an improper integral.
It might be helpful to think about infinite sums- absolutely convergent sums can be thought of as integrals of $L^1$ functions. Sums which are convergent but NOT absolutely convergent have peculiar properties. For example, take $\displaystyle \sum_{n=1}^{\infty} \frac{ (-1)^n }{n} $.
Although this sum has a classical limit, we may permute the terms in the sum to allow the limit to take any finite value. As a rough sketch of this process, we can consider the positive elements and the negative elements separately — each of these sums must diverge on their own, or the sum would converge absolutely.
Given any target sum $S$, we can take positive terms until the partial sum is greater than $S$, and then add negative terms until the partial sum is below $S$. Repeating in this fashion, the partial sums will oscillate around $S$, with the error term shrinking since the positive and negative terms tend to zero by convergence of the original sum.
Clearly there is something funny going on here: We can get a number for the sum of the series, but by rearranging we could get different numbers. In this same fashion we could construct functions with the same property — if we chop up the graph of the function and rearrange it, we can change the volume of the graph. This means that we can't really assign a meaningful notion of volume to the graph of a function if it isn't integrable in absolute value.