Example 7.3.1(d): Oddities of Riemann Integral
Can you take a Riemann integral over anything else but an interval?Yes and no. You can of course take Riemann integrals over unions of intervals, but nothing more complicated. The concept of a partition applies to an interval only, therefore the Riemann integral also works for intervals only.
That's odd: You can not integrate a function over, say, the Cantor middle third set. Since we know that many closed sets can be as complicated as the Cantor set, it seems that we exclude an aweful lot of sets.