Basics of Improper Riemann and Lebesgue-Integral

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?

• the intuition of improper integration is the concept of continuous extension of a function. That is: suppose that we define the function $F(x):=\int_a^xf(t)\mathrm dt$. Suppose that $F(c)$ doesnt exists but $\lim_{x\to c}F(x)$ exists. Then we can define $F(c):=\lim_{x\to c}F(x)$, then its said that $F$ was continuously extended to $c$. – Masacroso Jun 24 '17 at 8:31
• @Masacroso I am not clear.Plz explain In Lebesgue sense. – Subhajit Saha Jun 24 '17 at 9:33
• take a look here – Masacroso Jun 24 '17 at 9:36
• @Masacroso yes;f may be unbounded. But in cmpact domain I can make the overcome the probem I shall use the idea of continuous extension at the domain point for which f is unbounded by choosing a sufficiently large value of the function;The the area of the curve remains almost unaltered. Am I correct? – Subhajit Saha Jun 24 '17 at 10:30