I am interested in improper integration in both Riemann and Lebesgue sense.
In a compact set in the real line $[a,b]$ we partitioned $[a,b]$, then we calculate it Riemann sum. What was the approach or intuition to introduce improper Riemann integration over $[a,\infty)$? Because in this case I have no such partition.
In Lebesgue integration over a compact domain we take the partition on the range set of that function. But what should I do in the improper case? I believe that if a function $f$ is unbounded over both compact subset or rays in $\Bbb R$ then it is not Lebesgue integrable. Am I correct?