Interactive Real Analysis
- part of Example 7.3.1(e): Oddities of Riemann Integral
Could you define a Riemann integral of a function whose domain is not
R?
Back
Yes and no. Riemann integrals can certainly be defined for functions whose
domain are "intervals" in Rn. 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.
BackThat's odd: It is easy to define functions that have other spaces as their domain (sequences, for example, are functions from N to R). But the Riemann integrable does not lend itself to such functions.
Incidentally, wouldn't it be nice if we could say that if f is a function from N to R, i.e. f is a sequence { an }, thenThat way we could apply theorems for integrals to sums! But alas, that doesn't work for the Riemann integral ...f =
an