Interactive Real Analysis - part of

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Example 7.3.1(e): Oddities of Riemann Integral

Could you define a Riemann integral of a function whose domain is not R?
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.

That'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 }, then
f = an
That way we could apply theorems for integrals to sums! But alas, that doesn't work for the Riemann integral ...
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