## 7.3. Measures

### Example 7.3.1(e): Oddities of Riemann Integral

Yes and no. Riemann integrals can certainly be defined for functions whose domain are "intervals" in*. But a Riemann integral is dependent on partitions, which depend on the structure of the real line. Therefore, you can not define a Riemann integrable for functions defined on more abstract spaces.*

**R**^{n}
That's odd: It is easy to define functions that have other spaces as their
domain (sequences, for example, are functions from * N* to

*). But the Riemann integrable does not lend itself to such functions.*

**R**

Incidentally, wouldn't it be nice if we could say that iffis a function fromtoN, i.e.Rfis a sequence{ a, then_{n}}That way we could apply theorems for integrals to sums! But alas, that doesn't work for the Riemann integral ...f = a_{n}