Limit of the Riemann Integral

Question

A week ago my professor introduced us to the Riemann integral and said that we will use the graphic intuition that it calculates the area under a curve. He also said that the Riemann integral doesn't really define what the general area under a curve is and that we therefore need a theory of measurement (measure theory, I presume and as a generalisation of the Riemann integral the Lebesgue integral?). But I thought the whole point of the Riemann integral was to get a general definition for an area by using rectangles as approximations for the area because you can calculate those easily. What did I not understand? Why is measure theory necessary?

Thanks in advance.

Answer 1

The Lebesgue integrable functions on $[a,b]$ include the (proper) Riemann integrable functions and much much more. But the fact that we can integrate more functions is not the real reason measure theory is important. The real reason is it simply works better theoretically. For example "convergence theorems": If $f_n\to f$ pointwise and [some extra condition] then $\int_a^b f_n\to \int_a^b f$: With the Lebesgue integral we can give much better theorems of this form, with weaker extra conditions.

The Difference

An analogy, meant to illustrate the difference and why the Lebesgue integral "should" work better: Say you have a large nummber of people in a room. Each person has an age, which is a positive integer less than 100, and you want to find the sum of all the ages. Which procedure seems likely more efficient and accurate?

(i) You say $s=0$ at the start, go through the crowd one person at a time, adding each person's age to $s$.

(ii) You have everyone of age $1$ to raise their hands; you count, and record $n_1$ as the number of people aged $1$. Similarly you count $n_2$, the number of age $2$, etc., and then you calculate $n_1+2n_2+\dots+99 n_{99}$.

Seems clear to me that (ii) is a better idea. This is the difference between the two integrals; the Riemann integral uses an approximation analogous to (i), while the Lebesgue integral uses an approximation analogous to (ii).